@article{bgm22,
	abstract = {In this paper we propose the use of a continuous data assimilation algorithm for miscible flow models in a porous medium. In the absence of initial conditions for the model, observed sparse measurements are used to generate an approximation to the true solution. Under certain assumption of the sparse measurements and their incorporation into the algorithm it can be shown that the resulting approximate solution converges to the true solution at an exponential rate as time progresses. Various numerical examples are considered in order to validate the suitability of the algorithm.},
	author = {Bessaih, H. and Ginting, V. and McCaskill, B.},
	date = {2022/08/01},
	date-added = {2022-09-28 18:13:25 -0600},
	date-modified = {2022-09-28 18:13:25 -0600},
	doi = {10.1007/s00211-022-01306-y},
	id = {Bessaih2022},
	isbn = {0945-3245},
	journal = {Numerische Mathematik},
	number = {4},
	pages = {927--962},
	title = {Continuous {D}ata {A}ssimilation for {D}isplacement in a {P}orous {M}edium},
	url = {https://doi.org/10.1007/s00211-022-01306-y},
	volume = {151},
	year = {2022},
	bdsk-url-1 = {https://doi.org/10.1007/s00211-022-01306-y}}

In this paper we propose the use of a continuous data assimilation algorithm for miscible flow models in a porous medium. In the absence of initial conditions for the model, observed sparse measurements are used to generate an approximation to the true solution. Under certain assumption of the sparse measurements and their incorporation into the algorithm it can be shown that the resulting approximate solution converges to the true solution at an exponential rate as time progresses. Various numerical examples are considered in order to validate the suitability of the algorithm.

@article{ARYENI2022104199,
title = {A {S}emi-{A}nalytical {S}olution of {R}ichards {E}quation for {T}wo-{L}ayered {O}ne-{D}imensional {S}oil},
journal = {Advances in Water Resources},
volume = {165},
pages = {104199},
year = {2022},
issn = {0309-1708},
doi = {https://doi.org/10.1016/j.advwatres.2022.104199},
url = {https://www.sciencedirect.com/science/article/pii/S0309170822000719},
author = {T. Aryeni and V. Ginting},
keywords = {Richards Equation, Interface problem, Semi-analytical, Eigenfunction},
abstract = {A semi-analytical solution of the Richards Equation posed on a two-layered one-dimensional soil supplied with various boundary conditions is derived under a constraint that the constitutive relations are exponentially dependent on the pressure head. It allows for a transformation of the Richards Equation into a linear parabolic partial differential equation that governs a spatial–temporal function that represents the hydraulic conductivity. The procedure is proceeded with expressing this function as a linear combination of a set of eigenfunctions associated with a novel Sturm—Liouville problem that reflects the layer system and an auxiliary function that depends only on the spatial variable and the pressure head at the interface at the time of interest. All the relevant coefficients in the representation satisfy a nonlinear differential–algebraic system gathered from imposing the continuity of the pressure head and its flux at the interface. Two different approximations of the derivatives yield algebraic systems to be solved by the Newton method of iteration. Several pertinent numerical experiments demonstrating the approach are discussed and compared with the standard finite volume method.}
}

A semi-analytical solution of the Richards Equation posed on a two-layered one-dimensional soil supplied with various boundary conditions is derived under a constraint that the constitutive relations are exponentially dependent on the pressure head. It allows for a transformation of the Richards Equation into a linear parabolic partial differential equation that governs a spatial–temporal function that represents the hydraulic conductivity. The procedure is proceeded with expressing this function as a linear combination of a set of eigenfunctions associated with a novel Sturm—Liouville problem that reflects the layer system and an auxiliary function that depends only on the spatial variable and the pressure head at the interface at the time of interest. All the relevant coefficients in the representation satisfy a nonlinear differential–algebraic system gathered from imposing the continuity of the pressure head and its flux at the interface. Two different approximations of the derivatives yield algebraic systems to be solved by the Newton method of iteration. Several pertinent numerical experiments demonstrating the approach are discussed and compared with the standard finite volume method.

@article{adg22,
	abstract = {In this paper, we discuss an application of the generalized finite element method (GFEM) to approximate the solutions of quasilinear elliptic equations with multiple interfaces in one dimensional space. The problem is characterized by spatial discontinuity of the elliptic coefficient that depends on the unknown solution. It is known that unless the partition of the domain matches the discontinuity configuration, accuracy of standard finite element techniques significantly deteriorates and standard refinement of the partition may not suffice. The GFEM is a viable alternative to overcome this predicament. It is based on the construction of certain enrichment functions supplied to the standard finite element space that capture effects of the discontinuity. This approach is called stable (SGFEM) if it maintains an optimal rate of convergence and the conditioning of GFEM is not worse than that of the standard FEM. A convergence analysis is derived and performance of the method is illustrated by several numerical examples. Furthermore, it is known that typical global formulations such as FEMs do not enjoy the numerical local conservation property that is crucial in many conservation law-based applications. To remedy this issue, a Lagrange multiplier technique is adopted to enforce the local conservation. A numerical example is given to demonstrate the performance of proposed technique.},
	author = {Aryeni, T. and Deng, Q. and Ginting, V.},
	date = {2021/12/02},
	date-added = {2022-09-28 18:08:50 -0600},
	date-modified = {2022-09-28 18:08:50 -0600},
	doi = {10.1007/s10915-021-01675-w},
	id = {Aryeni2021},
	isbn = {1573-7691},
	journal = {Journal of Scientific Computing},
	number = {1},
	pages = {33},
	title = {On the {A}pplication of {S}table {G}eneralized {F}inite {E}lement {M}ethod for {Q}uasilinear {E}lliptic {T}wo-{P}oint {BVP}},
	url = {https://doi.org/10.1007/s10915-021-01675-w},
	volume = {90},
	year = {2021},
	bdsk-url-1 = {https://doi.org/10.1007/s10915-021-01675-w}}


%-------------------------2021-------------------------------

In this paper, we discuss an application of the generalized finite element method (GFEM) to approximate the solutions of quasilinear elliptic equations with multiple interfaces in one dimensional space. The problem is characterized by spatial discontinuity of the elliptic coefficient that depends on the unknown solution. It is known that unless the partition of the domain matches the discontinuity configuration, accuracy of standard finite element techniques significantly deteriorates and standard refinement of the partition may not suffice. The GFEM is a viable alternative to overcome this predicament. It is based on the construction of certain enrichment functions supplied to the standard finite element space that capture effects of the discontinuity. This approach is called stable (SGFEM) if it maintains an optimal rate of convergence and the conditioning of GFEM is not worse than that of the standard FEM. A convergence analysis is derived and performance of the method is illustrated by several numerical examples. Furthermore, it is known that typical global formulations such as FEMs do not enjoy the numerical local conservation property that is crucial in many conservation law-based applications. To remedy this issue, a Lagrange multiplier technique is adopted to enforce the local conservation. A numerical example is given to demonstrate the performance of proposed technique.

@article{1935-9179_2021_5_3405,
	title = {An {A}djoint-based {a} {P}osteriori {A}nalysis of {N}umerical {A}pproximation of {R}ichards {E}quation},
	journal = {Electronic Research Archive},
	volume = {29},
	number = {5},
	pages = {3405-3427},
	year = {2021},
	author = {Victor Ginting},
	abstract = {This paper formulates a general framework for a space-time finite element method for solving Richards Equation in one spatial dimension, where the spatial variable is discretized using the linear finite volume element and the temporal variable is discretized using a discontinuous Galerkin method. The actual implementation of a particular scheme is realized by imposing certain finite element space in temporal variable to the variational equation and appropriate "variational crime" in the form of numerical integrations for calculating integrations in the formulation. Once this is in place, adjoint-based error estimators for the approximate solution from the scheme is derived. The adjoint problem is obtained from an appropriate linearization of the nonlinear system. Numerical examples are presented to illustrate performance of the methods and the error estimators.}
}

%-------------------------2020-------------------------------

This paper formulates a general framework for a space-time finite element method for solving Richards Equation in one spatial dimension, where the spatial variable is discretized using the linear finite volume element and the temporal variable is discretized using a discontinuous Galerkin method. The actual implementation of a particular scheme is realized by imposing certain finite element space in temporal variable to the variational equation and appropriate "variational crime" in the form of numerical integrations for calculating integrations in the formulation. Once this is in place, adjoint-based error estimators for the approximate solution from the scheme is derived. The adjoint problem is obtained from an appropriate linearization of the nonlinear system. Numerical examples are presented to illustrate performance of the methods and the error estimators.

@article{ALMAMUN2020124980,
title = "Contaminant {T}ransport {F}orecasting in the {S}ubsurface using a {B}ayesian {F}ramework",
journal = "Applied Mathematics and Computation",
volume = "387",
pages = "124980",
year = "2020",
issn = "0096-3003",
doi = "https://doi.org/10.1016/j.amc.2019.124980",
url = "http://www.sciencedirect.com/science/article/pii/S0096300319309725",
author = "A. Al-Mamun and J. Barber and V. Ginting and F. Pereira and A. Rahunanthan",
keywords = "MCMC, Regularization, Two–stage proposal distribution, Uncertainty quantification, Convergence analysis, MPSRF",
abstract = "In monitoring subsurface aquifer contamination, we want to predict quantities—fractional flow curves of pollutant concentration—using subsurface fluid flow models with expertise and limited data. A Bayesian approach is considered here and the complexity associated with the simulation study presents an ongoing practical challenge. We use a Karhunen–Loève expansion for the permeability field in conjunction with GPU computing within a two–stage Markov Chain Monte Carlo (MCMC) method. Further reduction in computing costs is addressed by running several MCMC chains. We compare convergence criteria to quantify the uncertainty of predictions. Our contributions are two-fold: we first propose a fitting procedure for the Multivariate Potential Scale Reduction Factor (MPSRF) data that allows us to estimate the number of iterations for convergence. Then we present a careful analysis of ensembles of fractional flow curves suggesting that, for the problem at hand, the number of iterations required for convergence through the MPSRF analysis is excessive. Thus, for practical applications, our results provide an indication that an analysis of the posterior distributions of quantities of interest provides a reliable criterion to terminate MCMC simulations for quantifying uncertainty."
}

In monitoring subsurface aquifer contamination, we want to predict quantities—fractional flow curves of pollutant concentration—using subsurface fluid flow models with expertise and limited data. A Bayesian approach is considered here and the complexity associated with the simulation study presents an ongoing practical challenge. We use a Karhunen–Loève expansion for the permeability field in conjunction with GPU computing within a two–stage Markov Chain Monte Carlo (MCMC) method. Further reduction in computing costs is addressed by running several MCMC chains. We compare convergence criteria to quantify the uncertainty of predictions. Our contributions are two-fold: we first propose a fitting procedure for the Multivariate Potential Scale Reduction Factor (MPSRF) data that allows us to estimate the number of iterations for convergence. Then we present a careful analysis of ensembles of fractional flow curves suggesting that, for the problem at hand, the number of iterations required for convergence through the MPSRF analysis is excessive. Thus, for practical applications, our results provide an indication that an analysis of the posterior distributions of quantities of interest provides a reliable criterion to terminate MCMC simulations for quantifying uncertainty.

@article { ModifiedRadontransforminversionusingmoments,
      author = "Hayoung Choi and Victor Ginting and Farhad Jafari and Robert Mnatsakanov",
      title = "Modified {R}adon {T}ransform {I}nversion using {M}oments",
      journal = "Journal of Inverse and Ill-posed Problems",
      year = "2020",
      publisher = "De Gruyter",
      address = "Berlin, Boston",
      volume = "28",
      number = "1",
      doi = "https://doi.org/10.1515/jiip-2018-0090",
      pages=      "1-15",
      url = "https://www.degruyter.com/view/journals/jiip/28/1/article-p1.xml"
}

%-------------------------2019-------------------------------

@article{glifcaa19,
title = "{O}n the {F}ractional {D}iffusion-{A}dvection-{R}eaction {E}quation in $\mathbb{R}$",
journal = "Fractional Calculus \& Applied Analysis",
volume = "22",
number="4",
pages = "1039-1062",
year = "2019",
issn = "1311-0454",
author = "Victor Ginting and Yulong Li",
doi ="https://doi.org/10.1515/fca-2019-0055",
url ="https://www.degruyter.com/view/j/fca.2019.22.issue-4/fca-2019-0055/fca-2019-0055.xml?format=INT",
keywords = "Riemann-Liouville fractional operators, fractional diffusion, advection, reaction, weak fractional derivative, strong solution, regularity, infinite domain",
abstract = "We present an analysis of existence, uniqueness, and smoothness of the solution to a class of fractional ordinary differential equations posed on the whole real line that models a steady state behavior of a certain anomalous diffusion, advection, and reaction. The anomalous diffusion is modeled by the fractional Riemann-Liouville differential operators. The strong solution of the equation is sought in a Sobolev space defined by means of Fourier Transform. The key component of the analysis hinges on a characterization of this Sobolev space with the Riemann-Liouville derivatives that are understood in a weak sense. The existence, uniqueness, and smoothness of the solution is demonstrated with the assistance of several tools from functional and harmonic analyses."
}

We present an analysis of existence, uniqueness, and smoothness of the solution to a class of fractional ordinary differential equations posed on the whole real line that models a steady state behavior of a certain anomalous diffusion, advection, and reaction. The anomalous diffusion is modeled by the fractional Riemann-Liouville differential operators. The strong solution of the equation is sought in a Sobolev space defined by means of Fourier Transform. The key component of the analysis hinges on a characterization of this Sobolev space with the Riemann-Liouville derivatives that are understood in a weak sense. The existence, uniqueness, and smoothness of the solution is demonstrated with the assistance of several tools from functional and harmonic analyses.

@article{DENG2019166,
title = "{C}onstruction of {L}ocally {C}onservative {F}luxes for {H}igh {O}rder {C}ontinuous {G}alerkin {F}inite {E}lement {M}ethods",
journal = "Journal of Computational and Applied Mathematics",
volume = "359",
pages = "166-181",
year = "2019",
issn = "0377-0427",
doi = "https://doi.org/10.1016/j.cam.2019.03.049",
url = "http://www.sciencedirect.com/science/article/pii/S0377042719301803",
author = "Quanling Deng and Victor Ginting and Bradley McCaskill",
keywords = "CGFEM, FVEM, Conservative flux, Post-processing",
abstract = "Despite their robustness, it is known that standard continuous Galerkin Finite Element Methods (CGFEMs) do not produce a locally conservative flux field. As a result, their application to solving model problems that are derived from conservation laws can be limited. To remedy this issue some form of post-processing must be performed on the CGFEM solution. In this work, a simple post-processing technique is proposed to obtain a locally conservative flux field from a CGFEM solution. One distinct advantage of the proposed method is that it produces continuous normal flux at the element’s boundary. The post-processing is implemented on nodal-centered control volumes that are constructed from the original finite element mesh. The post-processing method is performed by solving an independent set of low dimensional problems posed on each element. The associated linear algebra systems are of dimension 12(k+1)(k+2) where k is the polynomial degree of CGFEM basis on a triangular mesh. A theoretical investigation is conducted to confirm that the post-processed solution converges in an optimal fashion to the true solution in the H1 semi-norm. Various numerical examples that demonstrate the performance of technique are given. Specifically, a simulation of a model for single-phase flow in a heterogeneous system is presented to show the necessity of the local conservation as well as the effective performance of the post-processing technique."
}

Despite their robustness, it is known that standard continuous Galerkin Finite Element Methods (CGFEMs) do not produce a locally conservative flux field. As a result, their application to solving model problems that are derived from conservation laws can be limited. To remedy this issue some form of post-processing must be performed on the CGFEM solution. In this work, a simple post-processing technique is proposed to obtain a locally conservative flux field from a CGFEM solution. One distinct advantage of the proposed method is that it produces continuous normal flux at the element’s boundary. The post-processing is implemented on nodal-centered control volumes that are constructed from the original finite element mesh. The post-processing method is performed by solving an independent set of low dimensional problems posed on each element. The associated linear algebra systems are of dimension 12(k+1)(k+2) where k is the polynomial degree of CGFEM basis on a triangular mesh. A theoretical investigation is conducted to confirm that the post-processed solution converges in an optimal fashion to the true solution in the H1 semi-norm. Various numerical examples that demonstrate the performance of technique are given. Specifically, a simulation of a model for single-phase flow in a heterogeneous system is presented to show the necessity of the local conservation as well as the effective performance of the post-processing technique.

@article{AKBARI201988,
title = "A {M}ultiscale {D}irect {S}olver for the {A}pproximation of {F}lows in {H}igh {C}ontrast {P}orous {M}edia",
journal = "Journal of Computational and Applied Mathematics",
volume = "359",
pages = "88-101",
year = "2019",
issn = "0377-0427",
doi = "https://doi.org/10.1016/j.cam.2019.03.028",
url = "http://www.sciencedirect.com/science/article/pii/S0377042719301530",
author = "Hani Akbari and Allan P. Engsig-Karup and Victor Ginting and Felipe Pereira",
keywords = "Elliptic equations, Combination algorithm, Multiscale method, Robin boundary condition, Parallel computing, High contrast media",
abstract = "We consider a non-overlapping domain decomposition approach to approximate the solution of elliptic boundary value problems with high contrast in their coefficients. We propose a method such that initially local solutions subject to Robin boundary conditions in each primal subdomain are constructed with (locally conservative) finite element or finite volume methods. Then, a novel approach is introduced to obtain a (discontinuous) global solution in terms of linear combination of the local subdomain solutions. In the proposed algorithm the computation of local solutions for unions of subdomains are localized at nearest-neighbor subdomain boundaries, thus avoiding the solution of global interface problems. We remove discontinuities in a smoothing step that is defined on a staggered grid or dual subdomains. The resulting algorithm is naturally parallelizable and can be employed as a parallel direct solver, offering great potential for the numerical solution of large problems. In fact, subdomains can be considered small enough to fit well in GPUs and the proposed procedure can handle adaptive (in space) simulations effectively. Numerical simulations are presented and discussed. We demonstrate the effectiveness of the proposed approach with two and three dimensional high contrast and channelized coefficients, that lead to challenging approximation problems. The new procedure, although designed for parallel processing, is also of value for serial calculations."
}

%-------------------------2018-------------------------------

We consider a non-overlapping domain decomposition approach to approximate the solution of elliptic boundary value problems with high contrast in their coefficients. We propose a method such that initially local solutions subject to Robin boundary conditions in each primal subdomain are constructed with (locally conservative) finite element or finite volume methods. Then, a novel approach is introduced to obtain a (discontinuous) global solution in terms of linear combination of the local subdomain solutions. In the proposed algorithm the computation of local solutions for unions of subdomains are localized at nearest-neighbor subdomain boundaries, thus avoiding the solution of global interface problems. We remove discontinuities in a smoothing step that is defined on a staggered grid or dual subdomains. The resulting algorithm is naturally parallelizable and can be employed as a parallel direct solver, offering great potential for the numerical solution of large problems. In fact, subdomains can be considered small enough to fit well in GPUs and the proposed procedure can handle adaptive (in space) simulations effectively. Numerical simulations are presented and discussed. We demonstrate the effectiveness of the proposed approach with two and three dimensional high contrast and channelized coefficients, that lead to challenging approximation problems. The new procedure, although designed for parallel processing, is also of value for serial calculations.

@article{WANG2018291,
title = "{A}nalysis of {N}onequilibrium {E}ffects and {F}low {I}nstability in {I}mmiscible {T}wo-{P}hase {F}low in {P}orous {M}edia",
journal = "Advances in Water Resources",
volume = "122",
pages = "291-303",
year = "2018",
issn = "0309-1708",
doi = "https://doi.org/10.1016/j.advwatres.2018.10.019",
url = "http://www.sciencedirect.com/science/article/pii/S0309170818303944",
author = "Yuhang Wang and Saman A. Aryana and Frederico Furtado and Victor Ginting",
keywords = "Multiphase flow, Nonequilibrium models, High-resolution methods, Flow instability",
abstract = "Two-dimensional, high-resolution, numerical solutions for the classical formulation and two widely accepted nonequilibrium models of multiphase flow through porous media are generated and compared with experimental observations from literature. Flow equations for simultaneous flow of two immiscible phases through porous media are written in a vorticity stream-function form. In the resulting system of equations, the vorticity stream-function equation is solved using a spectral method and the transport equation is discretized in space using a central-upwind scheme. The system of equations is solved for a two-dimensional domain using a semi-implicit time-stepper. The solutions reveal behavior that is not apparent in one-dimensional solutions, namely that sharpening of the saturation front caused by the inclusion of a dynamic capillary pressure results in propagation of viscous fingers compared to the classical formulation. The inclusion of nonequilibrium effects in the constitutive relations, in the form of effective saturation, introduces dispersion and smears the otherwise highly resolved viscous fingers in the saturation front. Once developed, the length of the mixing zone in the numerical solutions remains constant with time regardless of the degree of instability. This is contrary to the evolution of the mixing zone observed in unstable flow experiments where, unlike the numerical solutions, the propagation speed of the leading edge of the front appears to increase with time."
}

Two-dimensional, high-resolution, numerical solutions for the classical formulation and two widely accepted nonequilibrium models of multiphase flow through porous media are generated and compared with experimental observations from literature. Flow equations for simultaneous flow of two immiscible phases through porous media are written in a vorticity stream-function form. In the resulting system of equations, the vorticity stream-function equation is solved using a spectral method and the transport equation is discretized in space using a central-upwind scheme. The system of equations is solved for a two-dimensional domain using a semi-implicit time-stepper. The solutions reveal behavior that is not apparent in one-dimensional solutions, namely that sharpening of the saturation front caused by the inclusion of a dynamic capillary pressure results in propagation of viscous fingers compared to the classical formulation. The inclusion of nonequilibrium effects in the constitutive relations, in the form of effective saturation, introduces dispersion and smears the otherwise highly resolved viscous fingers in the saturation front. Once developed, the length of the mixing zone in the numerical solutions remains constant with time regardless of the degree of instability. This is contrary to the evolution of the mixing zone observed in unstable flow experiments where, unlike the numerical solutions, the propagation speed of the leading edge of the front appears to increase with time.

@article{ARYANA2018499,
title = "{O}n {S}eries {S}olution for {S}econd {O}rder {S}emilinear {P}arabolic {IBVP}s",
journal = "Journal of Computational and Applied Mathematics",
volume = "330",
pages = "499-518",
year = "2018",
issn = "0377-0427",
doi = "https://doi.org/10.1016/j.cam.2017.08.024",
url = "http://www.sciencedirect.com/science/article/pii/S0377042717304132",
author = "S. Aryana and F. Furtado and V. Ginting and P. Torsu",
keywords = "Adomian decomposition, Fourier series, Burgers equation, KPZ equation",
abstract = "This paper presents a method to obtain semi-analytical solutions to second order semilinear IBVPs where the standard Adomian decomposition may fail to yield nontrivial solutions or fail to produce the correct partial solutions. In contrast to the standard Adomian decomposition method, the proposed solution method is distinguished by simultaneous inversion of the linear differential operators using eigenfunctions expansion representations. The proposed method is applied to several examples of initial boundary value problems — a linear Advection–Diffusion problem, a Burgers equation and the deterministic Kardar–Parisi–Zhang (KPZ) equation. It is shown that the dependence of the series solution on the truncation order (N), the number of eigenfunctions (M) and the diffusion coefficient is rather complex."
}

This paper presents a method to obtain semi-analytical solutions to second order semilinear IBVPs where the standard Adomian decomposition may fail to yield nontrivial solutions or fail to produce the correct partial solutions. In contrast to the standard Adomian decomposition method, the proposed solution method is distinguished by simultaneous inversion of the linear differential operators using eigenfunctions expansion representations. The proposed method is applied to several examples of initial boundary value problems — a linear Advection–Diffusion problem, a Burgers equation and the deterministic Kardar–Parisi–Zhang (KPZ) equation. It is shown that the dependence of the series solution on the truncation order (N), the number of eigenfunctions (M) and the diffusion coefficient is rather complex.

@Article{Deng2018,
author="Deng, Quanling
and Ginting, Victor",
title="{L}ocally {C}onservative {C}ontinuous {G}alerkin {FEM} for {P}ressure {E}quation in {T}wo-{P}hase {F}low {M}odel in {S}ubsurfaces",
journal="Journal of Scientific Computing",
year="2018",
month="Mar",
day="01",
volume="74",
number="3",
pages="1264-1285",
abstract="A typical two-phase model for subsurface flow couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. In this paper, we study a combined method consisting of continuous Galerkin finite element methods (CGFEMs) followed by a post-processing technique for Darcy equation and a nodal centered finite volume method (FVM) with upwind schemes for the saturation transport equation, in which the coupled nonlinear problem is solved in the framework of operator decomposition. The post-processing technique is applied to CGFEM solutions to obtain locally conservative fluxes which ensures accuracy and robustness of the FVM solver for the saturation transport equation. We applied both upwind scheme and upwind scheme with slope limiter for FVM on triangular meshes in order to eliminate the non-physical oscillations. Various numerical examples are presented to demonstrate the performance of the overall methodology.",
issn="1573-7691",
doi="10.1007/s10915-017-0493-9",
url="https://doi.org/10.1007/s10915-017-0493-9"
}

%-------------------------2017-------------------------------

A typical two-phase model for subsurface flow couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. In this paper, we study a combined method consisting of continuous Galerkin finite element methods (CGFEMs) followed by a post-processing technique for Darcy equation and a nodal centered finite volume method (FVM) with upwind schemes for the saturation transport equation, in which the coupled nonlinear problem is solved in the framework of operator decomposition. The post-processing technique is applied to CGFEM solutions to obtain locally conservative fluxes which ensures accuracy and robustness of the FVM solver for the saturation transport equation. We applied both upwind scheme and upwind scheme with slope limiter for FVM on triangular meshes in order to eliminate the non-physical oscillations. Various numerical examples are presented to demonstrate the performance of the overall methodology.

@article{DENG201778,
title = "A locally conservative stabilized continuous {G}alerkin finite element method for two-phase flow in poroelastic subsurfaces",
journal = "Journal of Computational Physics",
volume = "347",
pages = "78-98",
year = "2017",
issn = "0021-9991",
doi = "https://doi.org/10.1016/j.jcp.2017.06.024",
url = "http://www.sciencedirect.com/science/article/pii/S0021999117304692",
author = "Q. Deng and V. Ginting and B. McCaskill and P. Torsu",
keywords = "Poroelastic, Geomechanic, Multiphase flow, Local conservation, Post-processing, Stabilized CGFEM",
abstract = "We study the application of a stabilized continuous Galerkin finite element method (CGFEM) in the simulation of multiphase flow in poroelastic subsurfaces. The system involves a nonlinear coupling between the fluid pressure, subsurface's deformation, and the fluid phase saturation, and as such, we represent this coupling through an iterative procedure. Spatial discretization of the poroelastic system employs the standard linear finite element in combination with a numerical diffusion term to maintain stability of the algebraic system. Furthermore, direct calculation of the normal velocities from pressure and deformation does not entail a locally conservative field. To alleviate this drawback, we propose an element based post-processing technique through which local conservation can be established. The performance of the method is validated through several examples illustrating the convergence of the method, the effectivity of the stabilization term, and the ability to achieve locally conservative normal velocities. Finally, the efficacy of the method is demonstrated through simulations of realistic multiphase flow in poroelastic subsurfaces."
}

%-------------------------2016-------------------------------

We study the application of a stabilized continuous Galerkin finite element method (CGFEM) in the simulation of multiphase flow in poroelastic subsurfaces. The system involves a nonlinear coupling between the fluid pressure, subsurface's deformation, and the fluid phase saturation, and as such, we represent this coupling through an iterative procedure. Spatial discretization of the poroelastic system employs the standard linear finite element in combination with a numerical diffusion term to maintain stability of the algebraic system. Furthermore, direct calculation of the normal velocities from pressure and deformation does not entail a locally conservative field. To alleviate this drawback, we propose an element based post-processing technique through which local conservation can be established. The performance of the method is validated through several examples illustrating the convergence of the method, the effectivity of the stabilization term, and the ability to achieve locally conservative normal velocities. Finally, the efficacy of the method is demonstrated through simulations of realistic multiphase flow in poroelastic subsurfaces.

@Article{Ginting2016,
author="Ginting, V. and Lin, G. and Liu, J.",
title="On {A}pplication of the {W}eak {G}alerkin {F}inite {E}lement {M}ethod to a {T}wo-{P}hase {M}odel for {S}ubsurface {F}low",
journal="Journal of Scientific Computing",
year="2016",
volume="66",
number="1",
pages="225-239",
abstract="This paper presents studies on applying the novel weak Galerkin finite element method (WGFEM) to a two-phase model for subsurface flow, which couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. The coupled problem is solved in the framework of operator decomposition. Specifically, the Darcy equation is solved by the WGFEM, whereas the saturation is solved by a finite volume method. The numerical velocity obtained from solving the Darcy equation by the WGFEM is locally conservative and has continuous normal components across element interfaces. This ensures accuracy and robustness of the finite volume solver for the saturation equation. Numerical experiments on benchmarks demonstrate that the combined methods can handle very well two-phase flow problems in high-contrast heterogeneous porous media.",
issn="1573-7691",
doi="10.1007/s10915-015-0021-8",
url="http://dx.doi.org/10.1007/s10915-015-0021-8"
}

%-------------------------2015-------------------------------

This paper presents studies on applying the novel weak Galerkin finite element method (WGFEM) to a two-phase model for subsurface flow, which couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. The coupled problem is solved in the framework of operator decomposition. Specifically, the Darcy equation is solved by the WGFEM, whereas the saturation is solved by a finite volume method. The numerical velocity obtained from solving the Darcy equation by the WGFEM is locally conservative and has continuous normal components across element interfaces. This ensures accuracy and robustness of the finite volume solver for the saturation equation. Numerical experiments on benchmarks demonstrate that the combined methods can handle very well two-phase flow problems in high-contrast heterogeneous porous media.

@article{Bush2015354,
title = "A {L}ocally {C}onservative {S}tress {R}ecovery {T}echnique for {C}ontinuous {G}alerkin {FEM} in {L}inear {E}lasticity ",
journal = "Computer Methods in Applied Mechanics and Engineering ",
volume = "286",
number = "",
pages = "354-372",
year = "2015",
note = "",
issn = "0045-7825",
doi = "http://dx.doi.org/10.1016/j.cma.2015.01.002",
url = "http://www.sciencedirect.com/science/article/pii/S0045782515000031",
author = "L. Bush and Q. Deng and V. Ginting",
keywords = "Stress recovery",
keywords = "Local conservation",
keywords = "Two-step postprocessing",
keywords = "Linear finite element",
keywords = "Displacement",
keywords = "Linear elasticity ",
abstract = "Abstract The standard continuous Galerkin finite element method (FEM) is a versatile and well understood method for solving partial differential equations. However, one shortcoming of the method is lack of continuity of derivatives of the approximate solution at element boundaries. This leads to undesirable consequences for a variety of problems such as a lack of local conservation. A two-step postprocessing technique is developed in order to obtain a local conservation from the standard continuous Galerkin FEM on a vertex centered dual mesh relative to the finite element mesh when applied to displacement based linear elasticity. The postprocessing requires an auxiliary fully Neumann problem to be solved on each finite element where local problems are independent of each other and involve solving two small linear algebra systems whose sizes are 3-by-3 when using linear finite elements on a triangular mesh for displacement based linear elasticity. The postprocessed stresses then satisfy local conservation on the dual mesh. An a priori error analysis and numerical simulations are provided. "
}

Abstract The standard continuous Galerkin finite element method (FEM) is a versatile and well understood method for solving partial differential equations. However, one shortcoming of the method is lack of continuity of derivatives of the approximate solution at element boundaries. This leads to undesirable consequences for a variety of problems such as a lack of local conservation. A two-step postprocessing technique is developed in order to obtain a local conservation from the standard continuous Galerkin FEM on a vertex centered dual mesh relative to the finite element mesh when applied to displacement based linear elasticity. The postprocessing requires an auxiliary fully Neumann problem to be solved on each finite element where local problems are independent of each other and involve solving two small linear algebra systems whose sizes are 3-by-3 when using linear finite elements on a triangular mesh for displacement based linear elasticity. The postprocessed stresses then satisfy local conservation on the dual mesh. An a priori error analysis and numerical simulations are provided.

@article{Chaudhry2015730,
title = "A {P}osteriori {E}rror {A}nalysis of {IMEX} {M}ulti-{S}tep {T}ime {I}ntegration {M}ethods for {A}dvection-{D}iffusion-{R}eaction {E}quations ",
journal = "Computer Methods in Applied Mechanics and Engineering ",
volume = "285",
number = "",
pages = "730-751",
year = "2015",
note = "",
issn = "0045-7825",
doi = "http://dx.doi.org/10.1016/j.cma.2014.11.015",
url = "http://www.sciencedirect.com/science/article/pii/S0045782514004411",
author = "J. Chaudhry and D. Estep and V. Ginting and J. Shadid and S. Tavener",
keywords = "Error estimation",
keywords = "Adjoint operator",
keywords = "Implicit–explicit schemes ",
abstract = "Implicit-Explicit (IMEX) schemes are an important and widely used class of time integration methods for both parabolic and hyperbolic partial differential equations. We develop accurate a posteriori error estimates for a user-defined quantity of interest for two classes of multi-step IMEX schemes for advection-diffusion-reaction problems. The analysis proceeds by recasting the IMEX schemes into a variational form suitable for a posteriori error analysis employing adjoint problems and computable residuals. The a posteriori estimates quantify distinct contributions from various aspects of the spatial and temporal discretizations, and can be used to evaluate discretization choices. Numerical results are presented that demonstrate the accuracy of the estimates for a representative set of problems. "
}

Implicit-Explicit (IMEX) schemes are an important and widely used class of time integration methods for both parabolic and hyperbolic partial differential equations. We develop accurate a posteriori error estimates for a user-defined quantity of interest for two classes of multi-step IMEX schemes for advection-diffusion-reaction problems. The analysis proceeds by recasting the IMEX schemes into a variational form suitable for a posteriori error analysis employing adjoint problems and computable residuals. The a posteriori estimates quantify distinct contributions from various aspects of the spatial and temporal discretizations, and can be used to evaluate discretization choices. Numerical results are presented that demonstrate the accuracy of the estimates for a representative set of problems.

@article {MR3354999,
    AUTHOR = {Chaudhry, J. and Estep, D. and Ginting, V. and Tavener, S.},
     TITLE = {A {P}osteriori {A}nalysis for {I}terative {S}olvers for {N}onautonomous
              {E}volution {P}roblems},
  JOURNAL = {SIAM/ASA Journal on Uncertainty Quantification},
    VOLUME = {3},
      YEAR = {2015},
    NUMBER = {1},
     PAGES = {434-459},
      ISSN = {2166-2525},
   MRCLASS = {65L05 (65L07 65L20 65L70)},
  MRNUMBER = {3354999},
       DOI = {10.1137/130949403},
       URL = {http://dx.doi.org/10.1137/130949403},
   ABSTRACT="We derive, implement, and test a posteriori error estimates for numerical methods for a nonautonomous linear system that involve iterative solution of the discrete equations. We consider two iterations: the Picard iteration and the Jacobi iteration for solving the discrete matrix-vector equations. To carry out the analysis, we define an appropriate adjoint problem for the numerical approximations using the matricant. We present a number of examples with interesting characteristics to illustrate the effectiveness of the estimate. We also present a comparison between the a posteriori error estimate and a conceptually simpler estimate obtained with a 'pseudoadjoint' problem."
}

We derive, implement, and test a posteriori error estimates for numerical methods for a nonautonomous linear system that involve iterative solution of the discrete equations. We consider two iterations: the Picard iteration and the Jacobi iteration for solving the discrete matrix-vector equations. To carry out the analysis, we define an appropriate adjoint problem for the numerical approximations using the matricant. We present a number of examples with interesting characteristics to illustrate the effectiveness of the estimate. We also present a comparison between the a posteriori error estimate and a conceptually simpler estimate obtained with a 'pseudoadjoint' problem.

@article {NUM:NUM21975,
author = {Deng, Q. and Ginting, V.},
title = {Construction of {L}ocally {C}onservative {F}luxes for the {SUPG} {M}ethod},
journal = {Numerical Methods for Partial Differential Equations},
volume = {31},
number = {6},
issn = {1098-2426},
url = {http://dx.doi.org/10.1002/num.21975},
doi = {10.1002/num.21975},
pages = {1971-1994},
keywords = {advection diffusion, advection dominated, CGFEM, conservative flux, postprocessing, SUPG},
year = {2015},
abstract="
We consider the construction of locally conservative fluxes by means of a simple postprocessing technique obtained from the finite element solutions of advection diffusion equations. It is known that a naive calculation of fluxes from these solutions yields nonconservative fluxes. We consider two finite element methods: the usual continuous Galerkin finite element method for solving nondominating advection diffusion equations and the streamline upwind/Petrov-Galerkin method for solving advection dominated problems. We then describe the postprocessing technique for constructing conservative fluxes from the numerical solutions of the general variational formulation. The postprocessing technique requires solving an auxiliary Neumann boundary value problem on each element independently and it produces a locally conservative flux on a vertex centered dual mesh relative to the finite element mesh. We provide a convergence analysis for the postprocessing technique. Performance of the technique and the convergence behavior are demonstrated through numerical examples including a set of test problems for advection diffusion equations, advection dominated equations, and drift-diffusion equations."
}

We consider the construction of locally conservative fluxes by means of a simple postprocessing technique obtained from the finite element solutions of advection diffusion equations. It is known that a naive calculation of fluxes from these solutions yields nonconservative fluxes. We consider two finite element methods: the usual continuous Galerkin finite element method for solving nondominating advection diffusion equations and the streamline upwind/Petrov-Galerkin method for solving advection dominated problems. We then describe the postprocessing technique for constructing conservative fluxes from the numerical solutions of the general variational formulation. The postprocessing technique requires solving an auxiliary Neumann boundary value problem on each element independently and it produces a locally conservative flux on a vertex centered dual mesh relative to the finite element mesh. We provide a convergence analysis for the postprocessing technique. Performance of the technique and the convergence behavior are demonstrated through numerical examples including a set of test problems for advection diffusion equations, advection dominated equations, and drift-diffusion equations.

@article{egh15numat,
year={2015},
issn={0029-599X},
journal={Numerische Mathematik},
volume={131},
number={4},
doi={10.1007/s00211-015-0703-z},
title={On {M}ultiscale {M}ethods in {P}etrov-{G}alerkin {F}ormulation},
url={http://dx.doi.org/10.1007/s00211-015-0703-z},
publisher={Springer Berlin Heidelberg},
keywords={35J15; 65N12; 65N30; 76S05},
author={Elfverson, D. and Ginting, V. and Henning, P.},
pages={643-682},
language={English},
abstract="In this work we investigate the advantages of multiscale methods in Petrov�Galerkin (PG) formulation in a general framework. The framework is based on a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space with good approximation properties and a high dimensional remainder space, which only contains negligible fine scale information. The multiscale space can then be used to obtain accurate Galerkin approximations. As a model problem we consider the Poisson equation. We prove that a Petrov-Galerkin formulation does not suffer from a significant loss of accuracy, and still preserve the convergence order of the original multiscale method. We also prove inf-sup stability of a PG continuous and a discontinuous Galerkin finite element multiscale method. Furthermore, we demonstrate that the Petrov-Galerkin method can decrease the computational complexity significantly, allowing for more efficient solution algorithms. As another application of the framework, we show how the Petrov-Galerkin framework can be used to construct a locally mass conservative solver for two-phase flow simulation that employs the Buckley�Leverett equation. To achieve this, we couple a PG discontinuous Galerkin finite element method with an upwind scheme for a hyperbolic conservation law."
}

In this work we investigate the advantages of multiscale methods in Petrov�Galerkin (PG) formulation in a general framework. The framework is based on a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space with good approximation properties and a high dimensional remainder space, which only contains negligible fine scale information. The multiscale space can then be used to obtain accurate Galerkin approximations. As a model problem we consider the Poisson equation. We prove that a Petrov-Galerkin formulation does not suffer from a significant loss of accuracy, and still preserve the convergence order of the original multiscale method. We also prove inf-sup stability of a PG continuous and a discontinuous Galerkin finite element multiscale method. Furthermore, we demonstrate that the Petrov-Galerkin method can decrease the computational complexity significantly, allowing for more efficient solution algorithms. As another application of the framework, we show how the Petrov-Galerkin framework can be used to construct a locally mass conservative solver for two-phase flow simulation that employs the Buckley�Leverett equation. To achieve this, we couple a PG discontinuous Galerkin finite element method with an upwind scheme for a hyperbolic conservation law.

@article{Ginting2015224,
title = "Multi-physics {M}arkov {C}hain {M}onte {C}arlo {M}ethods for {S}ubsurface {F}lows ",
journal = "Mathematics and Computers in Simulation ",
volume = "118",
number = "",
pages = "224-238",
year = "2015",
issn = "0378-4754",
doi = "http://dx.doi.org/10.1016/j.matcom.2014.11.023",
url = "http://www.sciencedirect.com/science/article/pii/S0378475414003231",
author = "V. Ginting and F. Pereira and A. Rahunanthan",
keywords = "MCMC",
keywords = "Porous media",
keywords = "Multi-physics",
keywords = "Multi-stage ",
abstract = "Abstract In \{CO\} 2 sequestration in deep saline aquifers, contaminant transport in subsurface, or oil or gas recovery, we often need to forecast flow patterns. In the flow forecasting, subsurface characterization is an important step. To characterize subsurface properties we establish a statistical description of the subsurface properties that are conditioned to existing dynamic (and static) data. We use a Markov chain Monte Carlo (MCMC) algorithm in a Bayesian statistical description to reconstruct the spatial distribution of two important subsurface properties: rock permeability and porosity. The \{MCMC\} algorithm requires repeatedly solving a set of nonlinear partial differential equations describing displacement of fluids in porous media for different values of permeability and porosity. The time needed for the generation of a reliable \{MCMC\} chain using the algorithm can be too long to be practical for flow forecasting. In this paper we develop computationally fast and effective methods of generating \{MCMC\} chains in the Bayesian framework for the subsurface characterization. Our strategy consists of constructing a family of computationally inexpensive preconditioners based on simpler physics as well as on surrogate models such that the number of fine-grid simulations is drastically reduced in the generation \{MCMC\} chains. We assess the quality of the proposed multi-physics \{MCMC\} methods by considering Monte Carlo simulations for forecasting oil production in an oil reservoir. "
}

Abstract In \CO\ 2 sequestration in deep saline aquifers, contaminant transport in subsurface, or oil or gas recovery, we often need to forecast flow patterns. In the flow forecasting, subsurface characterization is an important step. To characterize subsurface properties we establish a statistical description of the subsurface properties that are conditioned to existing dynamic (and static) data. We use a Markov chain Monte Carlo (MCMC) algorithm in a Bayesian statistical description to reconstruct the spatial distribution of two important subsurface properties: rock permeability and porosity. The \MCMC\ algorithm requires repeatedly solving a set of nonlinear partial differential equations describing displacement of fluids in porous media for different values of permeability and porosity. The time needed for the generation of a reliable \MCMC\ chain using the algorithm can be too long to be practical for flow forecasting. In this paper we develop computationally fast and effective methods of generating \MCMC\ chains in the Bayesian framework for the subsurface characterization. Our strategy consists of constructing a family of computationally inexpensive preconditioners based on simpler physics as well as on surrogate models such that the number of fine-grid simulations is drastically reduced in the generation \MCMC\ chains. We assess the quality of the proposed multi-physics \MCMC\ methods by considering Monte Carlo simulations for forecasting oil production in an oil reservoir.

@article{Johansson2015150,
title = "Adaptive {F}inite {E}lement {S}olution of {M}ultiscale {PDE}-{ODE} {S}ystems ",
journal = "Computer Methods in Applied Mechanics and Engineering ",
volume = "287",
number = "",
pages = "150-171",
year = "2015",
note = "",
issn = "0045-7825",
doi = "http://dx.doi.org/10.1016/j.cma.2015.01.010",
url = "http://www.sciencedirect.com/science/article/pii/S0045782515000237",
author = "A. Johansson and J. Chaudhry and V. Carey and D. Estep and V. Ginting and M. Larson and S. Tavener",
keywords = "A posteriori error analysis",
keywords = "Adaptive error control",
keywords = "Coupled physics",
keywords = "Multiscale model ",
abstract = "Abstract We consider adaptive finite element methods for a multiscale system consisting of a macroscale model comprising a system of reaction–diffusion partial differential equations coupled to a microscale model comprising a system of nonlinear ordinary differential equations. A motivating example is modeling the electrical activity of the heart taking into account the chemistry inside cells in the heart. Such multiscale models are computationally challenging due to the multiple scales in time and space that are involved. We describe a mathematically consistent approach to couple the microscale and macroscale models based on introducing an intermediate “coupling scale”. Since the ordinary differential equations are defined on a much finer spatial scale than the finite element discretization for the partial differential equation, we introduce a Monte Carlo approach to sampling the fine scale ordinary differential equations. We derive goal-oriented a posteriori error estimates for quantities of interest computed from the solution of the multiscale model using adjoint problems and computable residuals. We distinguish the errors in time and space for the partial differential equation and the ordinary differential equations separately and include errors due to the transfer of the solutions between the equations. The estimate also includes terms reflecting the sampling of the microscale model. Based on the accurate error estimates, we devise an adaptive solution method using a “blockwise” approach. The method and estimates are illustrated using a realistic problem. "
}

%-------------------------2014-------------------------------

Abstract We consider adaptive finite element methods for a multiscale system consisting of a macroscale model comprising a system of reaction–diffusion partial differential equations coupled to a microscale model comprising a system of nonlinear ordinary differential equations. A motivating example is modeling the electrical activity of the heart taking into account the chemistry inside cells in the heart. Such multiscale models are computationally challenging due to the multiple scales in time and space that are involved. We describe a mathematically consistent approach to couple the microscale and macroscale models based on introducing an intermediate “coupling scale”. Since the ordinary differential equations are defined on a much finer spatial scale than the finite element discretization for the partial differential equation, we introduce a Monte Carlo approach to sampling the fine scale ordinary differential equations. We derive goal-oriented a posteriori error estimates for quantities of interest computed from the solution of the multiscale model using adjoint problems and computable residuals. We distinguish the errors in time and space for the partial differential equation and the ordinary differential equations separately and include errors due to the transfer of the solutions between the equations. The estimate also includes terms reflecting the sampling of the microscale model. Based on the accurate error estimates, we devise an adaptive solution method using a “blockwise” approach. The method and estimates are illustrated using a realistic problem.

@article{Bi201478,
title = "Global {S}uperconvergence and {A} {P}osteriori {E}rror {E}stimates of the {F}inite {E}lement {M}ethod for {S}econd-{O}rder {Q}uasilinear {E}lliptic {P}roblems ",
journal = "Journal of Computational and Applied Mathematics ",
volume = "260",
number = "",
pages = "78-90",
year = "2014",
note = "",
issn = "0377-0427",
doi = "http://dx.doi.org/10.1016/j.cam.2013.09.042",
url = "http://www.sciencedirect.com/science/article/pii/S0377042713004974",
author = "C. Bi and V. Ginting",
keywords = "Quasi-linear elliptic problems",
keywords = "Finite element method",
keywords = "Superconvergence",
keywords = "Postprocessing-based a posteriori error estimates ",
abstract = "Abstract In this paper, we are concerned with the linear finite element approximations to the second-order quasi-linear elliptic problems. By means of an interpolation postprocessing technique, we develop the global superconvergence estimates in the H 1 - and W 1 , ∞ -norms provided the weak solutions are sufficiently smooth. Based on the global superconvergent approximations, we introduce and analyze the efficient postprocessing-based a posteriori error estimators, measured by the H 1 - and W 1 , ∞ -norms respectively. These can be used to assess the accuracy of the finite element solutions in applications. Numerical experiments are given to illustrate the global superconvergence estimates and the performance of the proposed estimators. "
}

Abstract In this paper, we are concerned with the linear finite element approximations to the second-order quasi-linear elliptic problems. By means of an interpolation postprocessing technique, we develop the global superconvergence estimates in the H 1 - and W 1 , ∞ -norms provided the weak solutions are sufficiently smooth. Based on the global superconvergent approximations, we introduce and analyze the efficient postprocessing-based a posteriori error estimators, measured by the H 1 - and W 1 , ∞ -norms respectively. These can be used to assess the accuracy of the finite element solutions in applications. Numerical experiments are given to illustrate the global superconvergence estimates and the performance of the proposed estimators.

@article{Bush2014395,
title = "Application of a {C}onservative, {G}eneralized {M}ultiscale {F}inite {E}lement {M}ethod to {F}low {M}odels ",
journal = "Journal of Computational and Applied Mathematics ",
volume = "260",
number = "",
pages = "395-409",
year = "2014",
note = "",
issn = "0377-0427",
doi = "http://dx.doi.org/10.1016/j.cam.2013.10.006",
url = "http://www.sciencedirect.com/science/article/pii/S0377042713005505",
author = "L. Bush and V. Ginting and M. Presho",
keywords = "Generalized multiscale finite element method",
keywords = "Flux conservation",
keywords = "Two-phase flow",
keywords = "Postprocessing ",
abstract = "Abstract In this paper, we propose a method for the construction of locally conservative flux fields from Generalized Multiscale Finite Element Method (GMsFEM) pressure solutions. The flux values are obtained from an element-based postprocessing procedure in which an independent set of 4 × 4 linear systems need to be solved. To test the performance of the method we consider two heterogeneous permeability coefficients and couple the resulting fluxes to a two-phase flow model. The increase in accuracy associated with the computation of the GMsFEM pressure solutions is inherited by the postprocessed flux fields and saturation solutions, and is closely correlated to the size of the reduced-order systems. In particular, the addition of more basis functions to the enriched coarse space yields solutions that more accurately capture the behavior of the fine scale model. A number of numerical examples are offered to validate the performance of the method. "
}

Abstract In this paper, we propose a method for the construction of locally conservative flux fields from Generalized Multiscale Finite Element Method (GMsFEM) pressure solutions. The flux values are obtained from an element-based postprocessing procedure in which an independent set of 4 × 4 linear systems need to be solved. To test the performance of the method we consider two heterogeneous permeability coefficients and couple the resulting fluxes to a two-phase flow model. The increase in accuracy associated with the computation of the GMsFEM pressure solutions is inherited by the postprocessed flux fields and saturation solutions, and is closely correlated to the size of the reduced-order systems. In particular, the addition of more basis functions to the enriched coarse space yields solutions that more accurately capture the behavior of the fine scale model. A number of numerical examples are offered to validate the performance of the method.

@article{Francisco2014125,
title = "Design and {I}mplementation of a {M}ultiscale {M}ixed {M}ethod {B}ased on a {N}onoverlapping {D}omain {D}ecomposition {P}rocedure ",
journal = "Mathematics and Computers in Simulation ",
volume = "99",
number = "",
pages = "125-138",
year = "2014",
issn = "0378-4754",
doi = "http://dx.doi.org/10.1016/j.matcom.2013.04.022",
url = "http://www.sciencedirect.com/science/article/pii/S0378475413001997",
author = "A. Francisco and V. Ginting and F. Pereira and J. Rigelo",
keywords = "Multiscale methods",
keywords = "Mixed finite elements",
keywords = "Domain decomposition ",
abstract = "We use a nonoverlapping iterative domain decomposition procedure based on the Robin interface condition to develop a new multiscale mixed method to compute the velocity field in heterogeneous porous media. Hybridized mixed finite elements are used for the spatial discretization of the equations. We define local, multiscale mixed basis functions to represent the discrete solutions in subdomains. Appropriate subspaces of the vector space spanned by these basis functions can be considered in the numerical approximations of heterogeneous porous media flow problems. The balance between numerical accuracy and numerical efficiency is determined by the choice of these subspaces. A detailed description of the numerical method is presented. Following that, numerical experiments are discussed to illustrate the important features of the new procedure and its comparison to the traditional fine grid simulations. "
}

We use a nonoverlapping iterative domain decomposition procedure based on the Robin interface condition to develop a new multiscale mixed method to compute the velocity field in heterogeneous porous media. Hybridized mixed finite elements are used for the spatial discretization of the equations. We define local, multiscale mixed basis functions to represent the discrete solutions in subdomains. Appropriate subspaces of the vector space spanned by these basis functions can be considered in the numerical approximations of heterogeneous porous media flow problems. The balance between numerical accuracy and numerical efficiency is determined by the choice of these subspaces. A detailed description of the numerical method is presented. Following that, numerical experiments are discussed to illustrate the important features of the new procedure and its comparison to the traditional fine grid simulations.

@article {MR3258565,
    AUTHOR = {Ginting, V. and Pereira, F. and Rahunanthan, A.},
     TITLE = {A {P}refetching {T}echnique for {P}rediction of {P}orous {M}edia {F}lows},
  JOURNAL = {Computational Geosciences},
    VOLUME = {18},
      YEAR = {2014},
    NUMBER = {5},
     PAGES = {661-675},
      ISSN = {1420-0597},
   MRCLASS = {76S05 (65C05)},
  MRNUMBER = {3258565},
       DOI = {10.1007/s10596-014-9413-3},
       URL = {http://dx.doi.org/10.1007/s10596-014-9413-3},
       ABSTRACT="In many applications in flows through porous media, one needs to determine the properties of subsurface to detect, monitor, or predict the actions of natural or induced forces. Here, we focus on two important subsurface properties: rock permeability and porosity. A Bayesian approach using a Markov Chain Monte Carlo (MCMC) algorithm is well suited for reconstructing the spatial distribution of permeability and porosity, and quantifying associated uncertainty in these properties. A crucial step in this approach is the computation of a likelihood function, which involves solving a possibly nonlinear system of partial differential equations. The computation time for the likelihood function limits the number of MCMC iterations that can be performed in a practical period of time. This affects the consistency of the posterior distribution of permeability and porosity obtained by MCMC exploration. To speed-up the posterior exploration, we can use a prefetching technique, which relies on the fact that multiple likelihoods of possible states into the future in an MCMC chain can be computed ahead of time. In this paper, we show that the prefetching technique implemented on multiple processors can make the Bayesian approach computationally tractable for subsurface characterization and prediction of porous media flows."
}

In many applications in flows through porous media, one needs to determine the properties of subsurface to detect, monitor, or predict the actions of natural or induced forces. Here, we focus on two important subsurface properties: rock permeability and porosity. A Bayesian approach using a Markov Chain Monte Carlo (MCMC) algorithm is well suited for reconstructing the spatial distribution of permeability and porosity, and quantifying associated uncertainty in these properties. A crucial step in this approach is the computation of a likelihood function, which involves solving a possibly nonlinear system of partial differential equations. The computation time for the likelihood function limits the number of MCMC iterations that can be performed in a practical period of time. This affects the consistency of the posterior distribution of permeability and porosity obtained by MCMC exploration. To speed-up the posterior exploration, we can use a prefetching technique, which relies on the fact that multiple likelihoods of possible states into the future in an MCMC chain can be computed ahead of time. In this paper, we show that the prefetching technique implemented on multiple processors can make the Bayesian approach computationally tractable for subsurface characterization and prediction of porous media flows.

@article{Ginting2014139,
title = "Rapid {Q}uantification of {U}ncertainty in {P}ermeability and {P}orosity of {O}il {R}eservoirs for {E}nabling {P}redictive {S}imulation ",
journal = "Mathematics and Computers in Simulation ",
volume = "99",
number = "",
pages = "139-152",
year = "2014",
issn = "0378-4754",
doi = "http://dx.doi.org/10.1016/j.matcom.2013.04.015",
url = "http://www.sciencedirect.com/science/article/pii/S0378475413000827",
author = "V. Ginting and F. Pereira and A. Rahunanthan",
keywords = "Dynamic data integration",
keywords = "GPU",
keywords = "Two-phase flows",
keywords = "Uncertainty quantification ",
abstract = "One of the most difficult tasks in subsurface flow simulations is the reliable characterization of properties of the subsurface. A typical situation employs dynamic data integration such as sparse (in space and time) measurements to be matched with simulated responses associated with a set of permeability and porosity fields. Among the challenges found in practice are proper mathematical modeling of the flow, persisting heterogeneity in the porosity and permeability, and the uncertainties inherent in them. In this paper we propose a Bayesian framework Monte Carlo Markov Chain (MCMC) simulation to sample a set of characteristics of the subsurface from the posterior distribution that are conditioned to the production data. This process requires obtaining the simulated responses over many realizations. In reality, this can be a prohibitively expensive endeavor with possibly many proposals rejection, and thus wasting the computational resources. To alleviate it, we employ a two-stage MCMC that includes a screening step of a proposal whose simulated response is obtained via an inexpensive coarse-scale model. A set of numerical examples using a two-phase flow problem in an oil reservoir as a benchmark application is given to illustrate the procedure and its use in predictive simulation. "
}




%-------------------------2013-------------------------------

One of the most difficult tasks in subsurface flow simulations is the reliable characterization of properties of the subsurface. A typical situation employs dynamic data integration such as sparse (in space and time) measurements to be matched with simulated responses associated with a set of permeability and porosity fields. Among the challenges found in practice are proper mathematical modeling of the flow, persisting heterogeneity in the porosity and permeability, and the uncertainties inherent in them. In this paper we propose a Bayesian framework Monte Carlo Markov Chain (MCMC) simulation to sample a set of characteristics of the subsurface from the posterior distribution that are conditioned to the production data. This process requires obtaining the simulated responses over many realizations. In reality, this can be a prohibitively expensive endeavor with possibly many proposals rejection, and thus wasting the computational resources. To alleviate it, we employ a two-stage MCMC that includes a screening step of a proposal whose simulated response is obtained via an inexpensive coarse-scale model. A set of numerical examples using a two-phase flow problem in an oil reservoir as a benchmark application is given to illustrate the procedure and its use in predictive simulation.

@article {MR3045707,
    AUTHOR = {Bi, C. and Ginting, V.},
     TITLE = {A {P}osteriori {E}rror {E}stimates of {D}iscontinuous {G}alerkin
              {M}ethod for {N}onmonotone {Q}uasi-{L}inear {E}lliptic {P}roblems},
  JOURNAL = {Journal of Scientific Computing},
    VOLUME = {55},
      YEAR = {2013},
    NUMBER = {3},
     PAGES = {659-687},
      ISSN = {0885-7474},
     CODEN = {JSCOEB},
   MRCLASS = {65N30 (65N15)},
  MRNUMBER = {3045707},
MRREVIEWER = {Igor Bock},
       DOI = {10.1007/s10915-012-9651-2},
       URL = {http://dx.doi.org/10.1007/s10915-012-9651-2},
 ABSTRACT="In this paper, we propose and study the residual-based a posteriori error estimates of h-version of symmetric interior penalty discontinuous Galerkin method for solving a class of second order quasi-linear elliptic problems which are of nonmonotone type. Computable upper and lower bounds on the error measured in terms of a natural mesh-dependent energy norm and the broken H 1-seminorm, respectively, are derived. Numerical experiments are also provided to illustrate the performance of the proposed estimators."
}

In this paper, we propose and study the residual-based a posteriori error estimates of h-version of symmetric interior penalty discontinuous Galerkin method for solving a class of second order quasi-linear elliptic problems which are of nonmonotone type. Computable upper and lower bounds on the error measured in terms of a natural mesh-dependent energy norm and the broken H 1-seminorm, respectively, are derived. Numerical experiments are also provided to illustrate the performance of the proposed estimators.

@article {MR3141756,
    AUTHOR = {Bush, L. and Ginting, V.},
     TITLE = {On the {A}pplication of the {C}ontinuous {G}alerkin {F}inite {E}lement
              {M}ethod for {C}onservation {P}roblems},
  JOURNAL = {SIAM Journal on Scientific Computing},
    VOLUME = {35},
      YEAR = {2013},
    NUMBER = {6},
     PAGES = {A2953-A2975},
      ISSN = {1064-8275},
   MRCLASS = {65N30 (65N15 65N50 65Y05)},
  MRNUMBER = {3141756},
MRREVIEWER = {Marius Ghergu},
       DOI = {10.1137/120900393},
       URL = {http://dx.doi.org/10.1137/120900393},
       ABSTRACT="
       One major drawback that prevents the use of the standard continuous Galerkin finite element method in solving conservation problems is its lack of a locally conservative flux. Our present work has developed a simple postprocessing for the continuous Galerkin finite element method resulting in a locally conservative flux on a vertex centered dual mesh relative to the finite element mesh. The postprocessing requires an auxiliary fully Neumann problem to be solved on each finite element. These local problems are independent of each other and in two dimensions involve solving only a 3-by-3 system in the case of triangular elements and a 4-by-4 system for quadrilateral elements. A convergence analysis for the method is provided and its performance is demonstrated through numerical examples of multiphase flow with triangular and quadrilateral elements along with a description of its parallel implementation."
}

One major drawback that prevents the use of the standard continuous Galerkin finite element method in solving conservation problems is its lack of a locally conservative flux. Our present work has developed a simple postprocessing for the continuous Galerkin finite element method resulting in a locally conservative flux on a vertex centered dual mesh relative to the finite element mesh. The postprocessing requires an auxiliary fully Neumann problem to be solved on each finite element. These local problems are independent of each other and in two dimensions involve solving only a 3-by-3 system in the case of triangular elements and a 4-by-4 system for quadrilateral elements. A convergence analysis for the method is provided and its performance is demonstrated through numerical examples of multiphase flow with triangular and quadrilateral elements along with a description of its parallel implementation.

@article{Chaudhry20131,
title = "A {P}osteriori {A}nalysis of an {I}terative {M}ulti-{D}iscretization {M}ethod for {R}eaction-{D}iffusion {S}ystems ",
journal = "Computer Methods in Applied Mechanics and Engineering ",
volume = "267",
number = "",
pages = "1-22",
year = "2013",
note = "",
issn = "0045-7825",
doi = "10.1016/j.cma.2013.08.007",
url = "http://www.sciencedirect.com/science/article/pii/S0045782513002107",
author = "J. Chaudhry and D. Estep and V. Ginting and S. Tavener",
keywords = "Reaction-diffusion",
keywords = "A posteriori estimates",
keywords = "Discontinuous Galerkin method",
keywords = "Multirate method",
keywords = "Multi-scale discretization",
keywords = "Operator decomposition ",
abstract = "This paper is concerned with the accurate computational error estimation of numerical solutions of multi-scale, multi-physics systems of reaction-diffusion equations. Such systems can present significantly different temporal and spatial scales within the components of the model, indicating the use of independent discretizations for different components. However, multi-discretization can have significant effects on accuracy and stability. We perform an adjoint-based analysis to derive asymptotically accurate a posteriori error estimates for a user-defined quantity of interest. These estimates account for leading order contributions to the error arising from numerical solution of each component, an error due to incomplete iteration, an error due to linearization, and for errors arising due to the projection of solution components between different spatial meshes. Several numerical examples with various settings are given to demonstrate the performance of the error estimators. "
}

This paper is concerned with the accurate computational error estimation of numerical solutions of multi-scale, multi-physics systems of reaction-diffusion equations. Such systems can present significantly different temporal and spatial scales within the components of the model, indicating the use of independent discretizations for different components. However, multi-discretization can have significant effects on accuracy and stability. We perform an adjoint-based analysis to derive asymptotically accurate a posteriori error estimates for a user-defined quantity of interest. These estimates account for leading order contributions to the error arising from numerical solution of each component, an error due to incomplete iteration, an error due to linearization, and for errors arising due to the projection of solution components between different spatial meshes. Several numerical examples with various settings are given to demonstrate the performance of the error estimators.

@article {MR3133290,
    AUTHOR = {Ginting, V. and Pereira, F. and Rahunanthan, A.},
     TITLE = {A {M}ulti-{S}tage {B}ayesian {P}rediction {F}ramework for {S}ubsurface
              {F}lows},
  JOURNAL = {International Journal for Uncertainty Quantification},
    VOLUME = {3},
      YEAR = {2013},
    NUMBER = {6},
     PAGES = {499-522},
      ISSN = {2152-5080},
   MRCLASS = {Database Expansion Item},
  MRNUMBER = {3133290},
       DOI = {10.1615/Int.J.UncertaintyQuantification.2013005281},
       URL =
              {http://dx.doi.org/10.1615/Int.J.UncertaintyQuantification.2013005281},
   ABSTRACT="We are concerned with the development of computationally efficient procedures for subsurface flow prediction that relies on the characterization of subsurface formations given static (measured permeability and porosity at well locations) and dynamic (measured produced fluid properties at well locations) data. We describe a predictive procedure in a Bayesian framework, which uses a single-phase flow model for characterization aiming at making prediction for a two-phase flow model. The quality of the characterization of the underlying formations is accessed through the prediction of future fluid flow production."
}

%-------------------------2012-------------------------------

We are concerned with the development of computationally efficient procedures for subsurface flow prediction that relies on the characterization of subsurface formations given static (measured permeability and porosity at well locations) and dynamic (measured produced fluid properties at well locations) data. We describe a predictive procedure in a Bayesian framework, which uses a single-phase flow model for characterization aiming at making prediction for a two-phase flow model. The quality of the characterization of the underlying formations is accessed through the prediction of future fluid flow production.

@article{Estep201210,
title = "A Posteriori {A}nalysis of a {M}ultirate {N}umerical {M}ethod for {O}rdinary {D}ifferential {E}quations ",
journal = "Computer Methods in Applied Mechanics and Engineering ",
volume = "223-224",
number = "",
pages = "10-27",
year = "2012",
note = "",
issn = "0045-7825",
doi = "10.1016/j.cma.2012.02.021",
url = "http://www.sciencedirect.com/science/article/pii/S0045782512000631",
author = "D. Estep and V. Ginting and S. Tavener",
keywords = "Adjoint operator",
keywords = "A posteriori estimates",
keywords = "Discontinuous Galerkin method",
keywords = "Iterative method",
keywords = "Multirate method",
keywords = "Multiscale integration",
keywords = "Operator decomposition ",
abstract = "In this paper, we analyze a multirate time integration method for systems of ordinary differential equations that present significantly different scales within the components of the model. The main purpose of this paper is to present a hybrid a priori-a posteriori error analysis that accounts for the effects of projections between the coarse and fine scale discretizations, the use of only a finite number of iterations in the iterative solution of the discrete equations, the numerical error arising in the solution of each component, and the effects on stability arising from the multirate solution. The hybrid estimate has the form of a computable a posteriori leading order expression and a provably-higher order a priori expression. We support this estimate by an a priori convergence analysis. We present several examples illustrating the accuracy of multirate integration schemes and the accuracy of the a posteriori estimate. "
}

%-------------------------2011-------------------------------

In this paper, we analyze a multirate time integration method for systems of ordinary differential equations that present significantly different scales within the components of the model. The main purpose of this paper is to present a hybrid a priori-a posteriori error analysis that accounts for the effects of projections between the coarse and fine scale discretizations, the use of only a finite number of iterations in the iterative solution of the discrete equations, the numerical error arising in the solution of each component, and the effects on stability arising from the multirate solution. The hybrid estimate has the form of a computable a posteriori leading order expression and a provably-higher order a priori expression. We support this estimate by an a priori convergence analysis. We present several examples illustrating the accuracy of multirate integration schemes and the accuracy of the a posteriori estimate.

@article {MR2832790,
    AUTHOR = {Bi, C. and Ginting, V.},
     TITLE = {Finite-{V}olume-{E}lement {M}ethod for {S}econd-{O}rder {Q}uasilinear
              {E}lliptic {P}roblems},
  JOURNAL = {IMA Journal of Numerical Analysis},
    VOLUME = {31},
      YEAR = {2011},
    NUMBER = {3},
     PAGES = {1062-1089},
      ISSN = {0272-4979},
     CODEN = {IJNADN},
   MRCLASS = {65N08 (35J25 35J62 65N12 65N15 65N30)},
  MRNUMBER = {2832790 (2012j:65367)},
MRREVIEWER = {Veronika Sobot{\'{\i}}kov{\'a}},
       DOI = {10.1093/imanum/drq011},
       URL = {http://dx.doi.org/10.1093/imanum/drq011},
ABSTRACT="In this paper we consider the finite-volume-element method for general second-order quasilinear elliptic problems over a convex polygonal domain in the plane. Using reasonable assumptions, we show the existence and uniqueness of the finite-volume-element approximations. It is proved that the finite-volume-element approximations are convergent with Graphic, where r > 2, and Graphic in the H1-, W1, �- and L2-norms, respectively, for u ? W2, r(�) and u ? W2, �(�) ? W3, p(�), where p > 1. Moreover, the optimal-order error estimates in the W1, �- and L2-norms and an Graphic estimate in the L�-norm are derived under the assumption that u ? W2, �(�) ? H3(�). Numerical experiments are presented to confirm the estimates."
}

In this paper we consider the finite-volume-element method for general second-order quasilinear elliptic problems over a convex polygonal domain in the plane. Using reasonable assumptions, we show the existence and uniqueness of the finite-volume-element approximations. It is proved that the finite-volume-element approximations are convergent with Graphic, where r > 2, and Graphic in the H1-, W1, �- and L2-norms, respectively, for u ? W2, r(�) and u ? W2, �(�) ? W3, p(�), where p > 1. Moreover, the optimal-order error estimates in the W1, �- and L2-norms and an Graphic estimate in the L�-norm are derived under the assumption that u ? W2, �(�) ? H3(�). Numerical experiments are presented to confirm the estimates.

@article {MR2853153,
    AUTHOR = {Bi, C. and Ginting, V.},
     TITLE = {Two-{G}rid {D}iscontinuous {G}alerkin {M}ethod for {Q}uasi-{L}inear
              {E}lliptic {P}roblems},
  JOURNAL = {Journal of Scientific Computing},
    VOLUME = {49},
      YEAR = {2011},
    NUMBER = {3},
     PAGES = {311-331},
      ISSN = {0885-7474},
     CODEN = {JSCOEB},
   MRCLASS = {65N30 (65N12)},
  MRNUMBER = {2853153 (2012m:65406)},
MRREVIEWER = {Alexandre L. Madureira},
       DOI = {10.1007/s10915-011-9463-9},
       URL = {http://dx.doi.org/10.1007/s10915-011-9463-9},
       ABSTRACT="In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree <i>r</i>&ge1 for a class of quasi-linear elliptic problems in . We propose a two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasi-linear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasi-linear elliptic problem on a coarse space. Convergence estimates in a broken <i>H</i><sup>1</sup>-norm are derived to justify the efficiency of the proposed two-grid algorithm. Numerical experiments are provided to confirm our theoretical findings. As a byproduct of the technique used in the analysis, we derive the optimal pointwise error estimates of the SIPG method for the quasi-linear elliptic problems in \mathbb{R}<sup><i>d</i></sup>, <i>d</i>=2,3 and use it to establish the convergence of the two-grid method for problems in
       &Omega &subset \mathbb{R}<sup>3</sup>."
}

In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree r&ge1 for a class of quasi-linear elliptic problems in . We propose a two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasi-linear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasi-linear elliptic problem on a coarse space. Convergence estimates in a broken H¹-norm are derived to justify the efficiency of the proposed two-grid algorithm. Numerical experiments are provided to confirm our theoretical findings. As a byproduct of the technique used in the analysis, we derive the optimal pointwise error estimates of the SIPG method for the quasi-linear elliptic problems in ℝ^d, d=2,3 and use it to establish the convergence of the two-grid method for problems in &Omega &subset ℝ³.

@article {MR2870986,
    AUTHOR = {Furtado, F. and Ginting, V. and Pereira, F. and Presho, M.},
     TITLE = {Operator {S}plitting {M}ultiscale {F}inite {V}olume {E}lement {M}ethod for
              {T}wo-{P}hase {F}low with {C}apillary {P}ressure},
  JOURNAL = {Transport in Porous Media},
    VOLUME = {90},
      YEAR = {2011},
    NUMBER = {3},
     PAGES = {927-947},
      ISSN = {0169-3913},
     CODEN = {TPMEEI},
   MRCLASS = {76B45 (65M08 76M12 76T99)},
  MRNUMBER = {2870986 (2012j:76031)},
       DOI = {10.1007/s11242-011-9824-8},
       URL = {http://dx.doi.org/10.1007/s11242-011-9824-8},
  ABSTRACT="A numerical method used for solving a two-phase flow problem as found in typical oil recovery is investigated in the setting of physics-based two-level operator splitting. The governing equations involve an elliptic differential equation coupled with a parabolic convection-dominated equation which poses a severe restriction for obtaining fully implicit numerical solutions. Furthermore, strong heterogeneity of the porous medium over many length scales adds to the complications for effectively solving the system. One viable approach is to split the system into three sub-systems: the elliptic, the hyperbolic, and the parabolic equation, respectively. In doing so, we allow for the use of appropriate numerical discretization for each type of equation and the careful exchange of information between them. We propose to use the multiscale finite volume element method (MsFVEM) for the elliptic and parabolic equations, and a nonoscillatory difference scheme for the hyperbolic equation. Performance of this procedure is confirmed through several numerical experiments."
}
	

A numerical method used for solving a two-phase flow problem as found in typical oil recovery is investigated in the setting of physics-based two-level operator splitting. The governing equations involve an elliptic differential equation coupled with a parabolic convection-dominated equation which poses a severe restriction for obtaining fully implicit numerical solutions. Furthermore, strong heterogeneity of the porous medium over many length scales adds to the complications for effectively solving the system. One viable approach is to split the system into three sub-systems: the elliptic, the hyperbolic, and the parabolic equation, respectively. In doing so, we allow for the use of appropriate numerical discretization for each type of equation and the careful exchange of information between them. We propose to use the multiscale finite volume element method (MsFVEM) for the elliptic and parabolic equations, and a nonoscillatory difference scheme for the hyperbolic equation. Performance of this procedure is confirmed through several numerical experiments.

@article {MR2842719,
    AUTHOR = {Ginting, V. and Pereira, F. and Presho, M. and
              Wo, S.},
     TITLE = {Application of the {T}wo-{S}tage {M}arkov {C}hain {M}onte {C}arlo
              {M}ethod for {C}haracterization of {F}ractured {R}eservoirs using a
              {S}urrogate {F}low {M}odel},
  JOURNAL = {Computational Geosciences},
    VOLUME = {15},
      YEAR = {2011},
    NUMBER = {4},
     PAGES = {691-707},
      ISSN = {1420-0597},
   MRCLASS = {76S05 (65C05 65C40)},
  MRNUMBER = {2842719 (2012i:76095)},
       DOI = {10.1007/s10596-011-9236-4},
       URL = {http://dx.doi.org/10.1007/s10596-011-9236-4},
  ABSTRACT="In this paper, we develop a procedure for subsurface characterization of a fractured porous medium. The characterization involves sampling from a representation of a fracture�s permeability that has been suitably adjusted to the dynamic tracer cut measurement data. We propose to use a type of dual-porosity, dual-permeability model for tracer flow. This model is built into the Markov chain Monte Carlo (MCMC) method in which the permeability is sampled. The Bayesian statistical framework is used to set the acceptance criteria of these samples and is enforced through sampling from the posterior distribution of the permeability fields conditioned to dynamic tracer cut data. In order to get a sample from the distribution, we must solve a series of problems which requires a fine-scale solution of the dual model. As direct MCMC is a costly method with the possibility of a low acceptance rate, we introduce a two-stage MCMC alternative which requires a suitable coarse-scale solution method of the dual model. With this filtering process, we are able to decrease our computational time as well as increase the proposal acceptance rate. A number of numerical examples are presented to illustrate the performance of the method."
}

In this paper, we develop a procedure for subsurface characterization of a fractured porous medium. The characterization involves sampling from a representation of a fracture�s permeability that has been suitably adjusted to the dynamic tracer cut measurement data. We propose to use a type of dual-porosity, dual-permeability model for tracer flow. This model is built into the Markov chain Monte Carlo (MCMC) method in which the permeability is sampled. The Bayesian statistical framework is used to set the acceptance criteria of these samples and is enforced through sampling from the posterior distribution of the permeability fields conditioned to dynamic tracer cut data. In order to get a sample from the distribution, we must solve a series of problems which requires a fine-scale solution of the dual model. As direct MCMC is a costly method with the possibility of a low acceptance rate, we introduce a two-stage MCMC alternative which requires a suitable coarse-scale solution method of the dual model. With this filtering process, we are able to decrease our computational time as well as increase the proposal acceptance rate. A number of numerical examples are presented to illustrate the performance of the method.

@article{Presho2011326,
title = "Calibrated {D}ual {P}orosity, {D}ual {P}ermeability {M}odeling of {F}ractured {R}eservoirs ",
journal = "Journal of Petroleum Science and Engineering ",
volume = "77",
number = "3-4",
pages = "326-337",
year = "2011",
note = "",
issn = "0920-4105",
doi = "10.1016/j.petrol.2011.04.007",
url = "http://www.sciencedirect.com/science/article/pii/S0920410511000878",
author = "M. Presho and S. Wo and V. Ginting",
keywords = "Dual porosity",
keywords = "Dual permeability",
keywords = "Pressure equation",
keywords = "Tracer flow ",
abstract = "In this paper we address the calibration of a classical dual porosity, dual permeability model which accounts for differences in matrix and fracture parameters. Fine scale benchmark solutions are obtained and we perform a comparison between dual porosity, dual permeability model solutions corresponding to a variety of possible parameter values. In the process, we describe an efficient method for solving the dual model pressure equations. We discuss applicable calibrations used for matching the dual porosity, dual permeability model results with those from the fine model and present a number of numerical examples to illustrate their performance. "
}

%-------------------------2010-------------------------------

In this paper we address the calibration of a classical dual porosity, dual permeability model which accounts for differences in matrix and fracture parameters. Fine scale benchmark solutions are obtained and we perform a comparison between dual porosity, dual permeability model solutions corresponding to a variety of possible parameter values. In the process, we describe an efficient method for solving the dual model pressure equations. We discuss applicable calibrations used for matching the dual porosity, dual permeability model results with those from the fine model and present a number of numerical examples to illustrate their performance.

@article{deeglcj10,
year={2010},
issn={1432-9360},
journal={Computing and Visualization in Science},
volume={13},
number={8},
doi={10.1007/s00791-011-0154-8},
title={Least {S}quares {A}pproach for {I}nitial {D}ata {R}ecovery in {D}ynamic {D}ata-{D}riven {A}pplications {S}imulations},
url={http://dx.doi.org/10.1007/s00791-011-0154-8},
publisher={Springer-Verlag},
keywords={Initial data recovery; Dynamic data-driven applications simulations (DDDAS); Least squares; Parameters update},
author={Douglas, C. and Efendiev, Y. and Ewing, R. and Ginting, V. and Lazarov, R. and Cole, M. and Jones, G.},
pages={365-375},
language={English},
abstract="In this paper, we consider the initial data recovery and the solution update based on the local measured data that are acquired during simulations. Each time new data is obtained, the initial condition, which is a representation of the solution at a previous time step, is updated. The update is performed using the least squares approach. The objective function is set up based on both a measurement error as well as a penalization term that depends on the prior knowledge about the solution at previous time steps (or initial data). Various numerical examples are considered, where the penalization term is varied during the simulations. Numerical examples demonstrate that the predictions are more accurate if the initial data are updated during the simulations."
}

In this paper, we consider the initial data recovery and the solution update based on the local measured data that are acquired during simulations. Each time new data is obtained, the initial condition, which is a representation of the solution at a previous time step, is updated. The update is performed using the least squares approach. The objective function is set up based on both a measurement error as well as a penalization term that depends on the prior knowledge about the solution at previous time steps (or initial data). Various numerical examples are considered, where the penalization term is varied during the simulations. Numerical examples demonstrate that the predictions are more accurate if the initial data are updated during the simulations.

@article{dfgmpp10rmg,
title = "On the {D}evelopment of a {H}igh-{P}erformance {T}ool for the {S}imulation of $\text{CO}_2$ {I}njection into {D}eep {S}aline {A}quifers",
journal = "Rocky Mountain Geology",
volume = "45",
pages = "151-161",
year = "2010",
note = "",
doi = "10.2113/gsrocky.45.2.151",
author = "C. Douglas and F. Furtado and V. Ginting and M. Mendes and F. Pereira and M. Piri"
}

@article {MR2644320,
    AUTHOR = {Ginting, V. and M{\aa}lqvist, A. and Presho, M.},
     TITLE = {A {N}ovel {M}ethod for {S}olving {M}ultiscale {E}lliptic {P}roblems with
              {R}andomly {P}erturbed {D}ata},
   JOURNAL = {Multiscale Model. Simul.},
  FJOURNAL = {Multiscale Modeling \& Simulation. A SIAM Interdisciplinary
              Journal},
    VOLUME = {8},
      YEAR = {2010},
    NUMBER = {3},
     PAGES = {977-996},
      ISSN = {1540-3459},
   MRCLASS = {65N30 (65C05)},
  MRNUMBER = {2644320 (2011e:65270)},
       DOI = {10.1137/090771302},
       URL = {http://dx.doi.org/10.1137/090771302},
   ABSTRACT="We propose a method for efficient solution of elliptic problems with multiscale features and randomly perturbed coefficients. We use the multiscale finite element method (MsFEM) as a starting point and derive an algorithm for solving a large number of multiscale problems in parallel. The method is intended to be used within a Monte Carlo framework where solutions corresponding to samples of the randomly perturbed data need to be computed. We show that the proposed method converges to the MsFEM solution in the limit for each individual sample of the data. We also show that the complexity of the method is proportional to one solve using MsFEM (where the fine scale is resolved) plus <i>N</i> (number of samples) solves of linear systems on the coarse scale, as opposed to solving <i>N</i> problems using MsFEM. A set of numerical examples is presented to illustrate the theoretical findings."
}

We propose a method for efficient solution of elliptic problems with multiscale features and randomly perturbed coefficients. We use the multiscale finite element method (MsFEM) as a starting point and derive an algorithm for solving a large number of multiscale problems in parallel. The method is intended to be used within a Monte Carlo framework where solutions corresponding to samples of the randomly perturbed data need to be computed. We show that the proposed method converges to the MsFEM solution in the limit for each individual sample of the data. We also show that the complexity of the method is proportional to one solve using MsFEM (where the fine scale is resolved) plus N (number of samples) solves of linear systems on the coarse scale, as opposed to solving N problems using MsFEM. A set of numerical examples is presented to illustrate the theoretical findings.

@article{Jiang2010862,
title = "Analysis of {G}lobal {M}ultiscale {F}inite {E}lement {M}ethods for {W}ave {E}quations with {C}ontinuum {S}patial {S}cales ",
journal = "Applied Numerical Mathematics ",
volume = "60",
number = "8",
pages = "862-876",
year = "2010",
note = "",
issn = "0168-9274",
doi = "10.1016/j.apnum.2010.04.011",
url = "http://www.sciencedirect.com/science/article/pii/S0168927410000759",
author = "L. Jiang and Y. Efendiev and V. Ginting",
keywords = "Galerkin multiscale finite element",
keywords = "Continuum scales",
keywords = "Wave equations ",
abstract = "In this paper, we discuss a numerical multiscale approach for solving wave equations with heterogeneous coefficients. Our interest comes from geophysics applications and we assume that there is no scale separation with respect to spatial variables. To obtain the solution of these multiscale problems on a coarse grid, we compute global fields such that the solution smoothly depends on these fields. We present a Galerkin multiscale finite element method using the global information and provide a convergence analysis when applied to solve the wave equations. We investigate the relation between the smoothness of the global fields and convergence rates of the global Galerkin multiscale finite element method for the wave equations. Numerical examples demonstrate that the use of global information renders better accuracy for wave equations with heterogeneous coefficients than the local multiscale finite element method. "
}		

In this paper, we discuss a numerical multiscale approach for solving wave equations with heterogeneous coefficients. Our interest comes from geophysics applications and we assume that there is no scale separation with respect to spatial variables. To obtain the solution of these multiscale problems on a coarse grid, we compute global fields such that the solution smoothly depends on these fields. We present a Galerkin multiscale finite element method using the global information and provide a convergence analysis when applied to solve the wave equations. We investigate the relation between the smoothness of the global fields and convergence rates of the global Galerkin multiscale finite element method for the wave equations. Numerical examples demonstrate that the use of global information renders better accuracy for wave equations with heterogeneous coefficients than the local multiscale finite element method.

@article{Presho20101130,
title = "Density {E}stimation of {T}wo-{P}hase {F}low with {M}ultiscale and {R}andomly {P}erturbed {D}ata ",
journal = "Advances in Water Resources ",
volume = "33",
number = "9",
pages = "1130-1141",
year = "2010",
note = "",
issn = "0309-1708",
doi = "10.1016/j.advwatres.2010.07.001",
url = "http://www.sciencedirect.com/science/article/pii/S0309170810001302",
author = "M. Presho and A. M{\aa}lqvist and V. Ginting",
keywords = "Multiscale finite element method",
keywords = "Elliptic equation",
keywords = "Random perturbation",
keywords = "Neumann series",
keywords = "Non-parametric density estimation",
keywords = "Two-phase flow ",
abstract = "In this paper, we describe an efficient approach for quantifying uncertainty in two-phase flow applications due to perturbations of the permeability in a multiscale heterogeneous porous medium. The method is based on the application of the multiscale finite element method within the framework of Monte Carlo simulation and an efficient preprocessing construction of the multiscale basis functions. The quantities of interest for our applications are the Darcy velocity and breakthrough time and we quantify their uncertainty by constructing the respective cumulative distribution functions. For the Darcy velocity we use the multiscale finite element method, but due to lack of conservation, we apply the multiscale finite volume element method as an alternative for use with the two-phase flow problem. We provide a number of numerical examples to illustrate the performance of the method. "
}

%-------------------------2009-------------------------------
		

In this paper, we describe an efficient approach for quantifying uncertainty in two-phase flow applications due to perturbations of the permeability in a multiscale heterogeneous porous medium. The method is based on the application of the multiscale finite element method within the framework of Monte Carlo simulation and an efficient preprocessing construction of the multiscale basis functions. The quantities of interest for our applications are the Darcy velocity and breakthrough time and we quantify their uncertainty by constructing the respective cumulative distribution functions. For the Darcy velocity we use the multiscale finite element method, but due to lack of conservation, we apply the multiscale finite volume element method as an alternative for use with the two-phase flow problem. We provide a number of numerical examples to illustrate the performance of the method.

@article {MR2557871,
    AUTHOR = {Bi, C. and Ginting, V.},
     TITLE = {A {R}esidual-{T}ype {A} {P}osteriori {E}rror {E}stimate of {F}inite {V}olume
              {E}lement {M}ethod for a {Q}uasi-{L}inear {E}lliptic {P}roblem},
   JOURNAL = {Numer. Math.},
  FJOURNAL = {Numerische Mathematik},
    VOLUME = {114},
      YEAR = {2009},
    NUMBER = {1},
     PAGES = {107-132},
      ISSN = {0029-599X},
     CODEN = {NUMMA7},
   MRCLASS = {65N08 (35J62 65N15)},
  MRNUMBER = {2557871 (2011k:65147)},
MRREVIEWER = {Pascal Omnes},
       DOI = {10.1007/s00211-009-0247-1},
       URL = {http://dx.doi.org/10.1007/s00211-009-0247-1},
 ABSTRACT="In this paper, we analyze a residual-type a posteriori error estimator of the finite volume element method for a quasi-linear elliptic problem of nonmonotone type and derive computable upper and lower bounds on the error in the <i>H</i><sup>1</sup>-norm. Numerical experiments are provided to illustrate the performance of the proposed estimator."
}
		

In this paper, we analyze a residual-type a posteriori error estimator of the finite volume element method for a quasi-linear elliptic problem of nonmonotone type and derive computable upper and lower bounds on the error in the H¹-norm. Numerical experiments are provided to illustrate the performance of the proposed estimator.

@article {MR2559634,
    AUTHOR = {Presho, M. and Ginting, V. and Wo, S.},
     TITLE = {The {U}ncertainty {E}ffects of {D}eformation {B}ands on {F}luid {F}low},
  JOURNAL = {Computational \& Applied Mathematics},
    VOLUME = {28},
      YEAR = {2009},
    NUMBER = {3},
     PAGES = {291-308},
      ISSN = {1807-0302},
   MRCLASS = {65C05 (76B10)},
  MRNUMBER = {2559634 (2010k:65009)},
}

%-------------------------2008-------------------------------

@article{Chu2008599,
title = "Flow {B}ased {O}versampling {T}echnique for {M}ultiscale {F}inite {E}lement {M}ethods ",
journal = "Advances in Water Resources ",
volume = "31",
number = "4",
pages = "599-608",
year = "2008",
note = "",
issn = "0309-1708",
doi = "10.1016/j.advwatres.2007.11.005",
url = "http://www.sciencedirect.com/science/article/pii/S030917080700173X",
author = "J. Chu and Y. Efendiev and V. Ginting and T. Hou",
keywords = "Multiscale",
keywords = "Finite volume",
keywords = "Oversampling",
keywords = "Upscaling",
keywords = "Two-phase flow ",
abstract = "Oversampling techniques are often used in porous media simulations to achieve high accuracy in multiscale simulations. These methods reduce the effect of artificial boundary conditions that are imposed in computing local quantities, such as upscaled permeabilities or basis functions. In the problems without scale separation and strong non-local effects, the oversampling region is taken to be the entire domain. The basis functions are computed using single-phase flow solutions which are further used in dynamic two-phase simulations. The standard oversampling approaches employ generic global boundary conditions which are not associated with actual flow boundary conditions. In this paper, we propose a flow based oversampling method where the actual two-phase flow boundary conditions are used in constructing oversampling auxiliary functions. Our numerical results show that the flow based oversampling approach is several times more accurate than the standard oversampling method. We provide partial theoretical explanation for these numerical observations. "
}

Oversampling techniques are often used in porous media simulations to achieve high accuracy in multiscale simulations. These methods reduce the effect of artificial boundary conditions that are imposed in computing local quantities, such as upscaled permeabilities or basis functions. In the problems without scale separation and strong non-local effects, the oversampling region is taken to be the entire domain. The basis functions are computed using single-phase flow solutions which are further used in dynamic two-phase simulations. The standard oversampling approaches employ generic global boundary conditions which are not associated with actual flow boundary conditions. In this paper, we propose a flow based oversampling method where the actual two-phase flow boundary conditions are used in constructing oversampling auxiliary functions. Our numerical results show that the flow based oversampling approach is several times more accurate than the standard oversampling method. We provide partial theoretical explanation for these numerical observations.

@article{1742-6596-125-1-012075,
  author={D. Estep and V. Carey and V. Ginting and S. Tavener and T. Wildey},
  title={A {P}osteriori {E}rror {A}nalysis of {M}ultiscale {O}perator {D}ecomposition {M}ethods for {M}ultiphysics {M}odels},
  journal={Journal of Physics: Conference Series},
  volume={125},
  number={1},
  pages={012075},
  url={http://stacks.iop.org/1742-6596/125/i=1/a=012075},
  year={2008},
  doi={10.1088/1742-6596/125/1/012075},
  abstract="Multiphysics, multiscale models present significant challenges in computing accurate solutions and for estimating the error in information computed from numerical solutions. In this paper, we describe recent advances in extending the techniques of a posteriori error analysis to multiscale operator decomposition solution methods. While the particulars of the analysis vary considerably with the problem, several key ideas underlie a general approach being developed to treat operator decomposition multiscale methods. We explain these ideas in the context of three specific examples."
}
	

Multiphysics, multiscale models present significant challenges in computing accurate solutions and for estimating the error in information computed from numerical solutions. In this paper, we describe recent advances in extending the techniques of a posteriori error analysis to multiscale operator decomposition solution methods. While the particulars of the analysis vary considerably with the problem, several key ideas underlie a general approach being developed to treat operator decomposition multiscale methods. We explain these ideas in the context of three specific examples.

@article {MR2390987,
    AUTHOR = {Estep, D. and Ginting, V. and Ropp, D. and Shadid, J. and Tavener, S.},
     TITLE = {An {A} {P}osteriori-{A} {P}riori {A}nalysis of {M}ultiscale {O}perator
              {S}plitting},
  JOURNAL = {SIAM Journal on Numerical Analysis},
    VOLUME = {46},
      YEAR = {2008},
    NUMBER = {3},
     PAGES = {1116-1146},
      ISSN = {0036-1429},
   MRCLASS = {65M20},
  MRNUMBER = {2390987 (2009i:65156)},
MRREVIEWER = {S{\"o}ren Bartels},
       DOI = {10.1137/07068237X},
       URL = {http://dx.doi.org/10.1137/07068237X},
 ABSTRACT="In this paper, we analyze a multiscale operator splitting method for solving systems of ordinary differential equations such as those that result upon space discretization of a reaction-diffusion equation. Our goal is to analyze and accurately estimate the error of the numerical solution, including the effects of any instabilities that can result from multiscale operator splitting. We present both an a priori error analysis and a new type of hybrid a priori-a posteriori error analysis for an operator splitting discontinuous Galerkin finite element method. Both analyses clearly distinguish between the effects of the operator splitting and the discretization of each component of the decomposed problem. The hybrid analysis has the form of a computable a posteriori leading order expression and a provably higher order a priori expression. The hybrid analysis takes into account the fact that the adjoint problems for the original problem and a multiscale operator splitting discretization differ in significant ways. In particular, this provides the means to monitor global instabilities that can arise from operator splitting."
}

%-------------------------2007-------------------------------
		

In this paper, we analyze a multiscale operator splitting method for solving systems of ordinary differential equations such as those that result upon space discretization of a reaction-diffusion equation. Our goal is to analyze and accurately estimate the error of the numerical solution, including the effects of any instabilities that can result from multiscale operator splitting. We present both an a priori error analysis and a new type of hybrid a priori-a posteriori error analysis for an operator splitting discontinuous Galerkin finite element method. Both analyses clearly distinguish between the effects of the operator splitting and the discretization of each component of the decomposed problem. The hybrid analysis has the form of a computable a posteriori leading order expression and a provably higher order a priori expression. The hybrid analysis takes into account the fact that the adjoint problems for the original problem and a multiscale operator splitting discretization differ in significant ways. In particular, this provides the means to monitor global instabilities that can arise from operator splitting.

@article {MR2358002,
    AUTHOR = {Bi, C. and Ginting, V.},
     TITLE = {Two-{G}rid {F}inite {V}olume {E}lement {M}ethod for {L}inear and {N}onlinear
              {E}lliptic {P}roblems},
  JOURNAL = {Numerische Mathematik},
    VOLUME = {108},
      YEAR = {2007},
    NUMBER = {2},
     PAGES = {177-198},
      ISSN = {0029-599X},
     CODEN = {NUMMA7},
   MRCLASS = {65N06},
  MRNUMBER = {2358002 (2008i:65227)},
MRREVIEWER = {Zheng Hui Xie},
       DOI = {10.1007/s00211-007-0115-9},
       URL = {http://dx.doi.org/10.1007/s00211-007-0115-9},
 ABSTRACT="Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse and one fine grid, are presented for the two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems and the two-dimensional second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms. A set of numerical examples are presented to confirm the estimates."
}

Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse and one fine grid, are presented for the two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems and the two-dimensional second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms. A set of numerical examples are presented to confirm the estimates.

@article{Durlofsky2007576,
title = "An {A}daptive {L}ocal-{G}lobal {M}ultiscale {F}inite {V}olume {E}lement {M}ethod for {T}wo-{P}hase {F}low {S}imulations ",
journal = "Advances in Water Resources ",
volume = "30",
number = "3",
pages = "576-588",
year = "2007",
note = "",
issn = "0309-1708",
doi = "10.1016/j.advwatres.2006.04.002",
url = "http://www.sciencedirect.com/science/article/pii/S0309170806000650",
author = "L. Durlofsky and Y. Efendiev and V. Ginting",
keywords = "Subsurface",
keywords = "Flow simulation",
keywords = "Heterogeneity",
keywords = "Multiscale",
keywords = "Upscaling",
keywords = "Finite element",
keywords = "Finite volume",
keywords = "Subgrid",
keywords = "Transport",
keywords = "Local–global ",
abstract = "Multiscale solution methods are currently under active investigation for the simulation of subsurface flow in heterogeneous formations. These procedures capture the effects of fine scale permeability variations through the calculation of specialized coarse scale basis functions. Most of the multiscale techniques presented to date employ localization approximations in the calculation of these basis functions. For some highly correlated (e.g., channelized) formations, however, global effects are important and these may need to be incorporated into the multiscale basis functions. This can be accomplished using global fine scale simulations, but this may be computationally expensive. In this paper an adaptive local–global technique, originally developed within the context of upscaling, is applied for the computation of multiscale basis functions. The procedure enables the efficient incorporation of approximate global information, determined via coarse scale simulations, into the multiscale basis functions. The resulting procedure is formulated as a finite volume element method and is applied for a number of single- and two-phase flow simulations of channelized two-dimensional systems. Both conforming and nonconforming procedures are considered. The level of accuracy of the resulting method is shown to be consistently higher than that of the standard finite volume element multiscale technique based on localized basis functions determined using linear pressure boundary conditions. "
}
		

Multiscale solution methods are currently under active investigation for the simulation of subsurface flow in heterogeneous formations. These procedures capture the effects of fine scale permeability variations through the calculation of specialized coarse scale basis functions. Most of the multiscale techniques presented to date employ localization approximations in the calculation of these basis functions. For some highly correlated (e.g., channelized) formations, however, global effects are important and these may need to be incorporated into the multiscale basis functions. This can be accomplished using global fine scale simulations, but this may be computationally expensive. In this paper an adaptive local–global technique, originally developed within the context of upscaling, is applied for the computation of multiscale basis functions. The procedure enables the efficient incorporation of approximate global information, determined via coarse scale simulations, into the multiscale basis functions. The resulting procedure is formulated as a finite volume element method and is applied for a number of single- and two-phase flow simulations of channelized two-dimensional systems. Both conforming and nonconforming procedures are considered. The level of accuracy of the resulting method is shown to be consistently higher than that of the standard finite volume element multiscale technique based on localized basis functions determined using linear pressure boundary conditions.

@article {MR2342125,
    AUTHOR = {Jiang, L. and Efendiev, Y. and Ginting, V.},
     TITLE = {Multiscale {M}ethods for {P}arabolic {E}quations with {C}ontinuum
              {S}patial {S}cales},
  JOURNAL = {Discrete and Continuous Dynamical Systems. Series B. A Journal
              Bridging Mathematics and Sciences},
    VOLUME = {8},
      YEAR = {2007},
    NUMBER = {4},
     PAGES = {833-859},
      ISSN = {1531-3492},
   MRCLASS = {65M99 (34G10 35K20)},
  MRNUMBER = {2342125 (2008k:65216)},
MRREVIEWER = {Karsten Urban},
       DOI = {10.3934/dcdsb.2007.8.833},
       URL = {http://dx.doi.org/10.3934/dcdsb.2007.8.833},
       ABSTRACT="In this paper, we consider multiscale approaches for solving parabolic equations with heterogeneous coefficients. Our interest stems from porous media applications and we assume that there is no scale separation with respect to spatial variables. To compute the solution of these multiscale problems on a coarse grid, we define global fields such that the solution smoothly depends on these fields. We present various finite element discretization techniques and provide analyses of these methods. A few representative numerical examples are presented using heterogeneous fields with strong non-local features. These numerical results demonstrate that the solution can be captured more accurately on the coarse grid when some type of limited global information is used."
}

%-------------------------2006-------------------------------

In this paper, we consider multiscale approaches for solving parabolic equations with heterogeneous coefficients. Our interest stems from porous media applications and we assume that there is no scale separation with respect to spatial variables. To compute the solution of these multiscale problems on a coarse grid, we define global fields such that the solution smoothly depends on these fields. We present various finite element discretization techniques and provide analyses of these methods. A few representative numerical examples are presented using heterogeneous fields with strong non-local features. These numerical results demonstrate that the solution can be captured more accurately on the coarse grid when some type of limited global information is used.

@article{Efendiev2006155,
title = "Accurate {M}ultiscale {F}inite {E}lement {M}ethods for {T}wo-{P}hase {F}low {S}imulations ",
journal = "Journal of Computational Physics ",
volume = "220",
number = "1",
pages = "155-174",
year = "2006",
note = "",
issn = "0021-9991",
doi = "10.1016/j.jcp.2006.05.015",
url = "http://www.sciencedirect.com/science/article/pii/S0021999106002269",
author = "Y. Efendiev and V. Ginting and T. Hou and R. Ewing",
keywords = "Multiscale",
keywords = "Finite element",
keywords = "Finite volume",
keywords = "Global",
keywords = "Two-phase",
keywords = "Upscaling ",
abstract = "In this paper we propose a modified multiscale finite element method for two-phase flow simulations in heterogeneous porous media. The main idea of the method is to use the global fine-scale solution at initial time to determine the boundary conditions of the basis functions. This method provides a significant improvement in two-phase flow simulations in porous media where the long-range effects are important. This is typical for some recent benchmark tests, such as the \{SPE\} comparative solution project [M. Christie, M. Blunt, Tenth spe comparative solution project: a comparison of upscaling techniques, \{SPE\} Reser. Eval. Eng. 4 (2001) 308–317], where porous media have a channelized structure. The use of global information allows us to capture the long-range effects more accurately compared to the multiscale finite element methods that use only local information to construct the basis functions. We present some analysis of the proposed method to illustrate that the method can indeed capture the long-range effect in channelized media. "
}

In this paper we propose a modified multiscale finite element method for two-phase flow simulations in heterogeneous porous media. The main idea of the method is to use the global fine-scale solution at initial time to determine the boundary conditions of the basis functions. This method provides a significant improvement in two-phase flow simulations in porous media where the long-range effects are important. This is typical for some recent benchmark tests, such as the \SPE\ comparative solution project [M. Christie, M. Blunt, Tenth spe comparative solution project: a comparison of upscaling techniques, \SPE\ Reser. Eval. Eng. 4 (2001) 308–317], where porous media have a channelized structure. The use of global information allows us to capture the long-range effects more accurately compared to the multiscale finite element methods that use only local information to construct the basis functions. We present some analysis of the proposed method to illustrate that the method can indeed capture the long-range effect in channelized media.

@article{deegl07,
year={2006},
issn={0010-485X},
journal={Computing},
volume={77},
number={4},
doi={10.1007/s00607-006-0165-3},
title={Dynamic {D}ata {D}riven {S}imulations in {S}tochastic {E}nvironments},
url={http://dx.doi.org/10.1007/s00607-006-0165-3},
publisher={Springer-Verlag},
keywords={65N99; MCMC; porous media flow; uncertainty; permeability; DDDAS},
author={Douglas, C. and Efendiev, Y. and Ewing, R. and Ginting, V. and Lazarov, R.},
pages={321-333},
language={English},
abstract="To improve the predictions in dynamic data driven simulations (DDDAS) for subsurface problems, we propose the permeability update based on observed measurements. Based on measurement errors and a <i>priori</i> information about the permeability field, such as covariance of permeability field and its values at the measurement locations, the permeability field is sampled. This sampling problem is highly nonlinear and Markov chain Monte Carlo (MCMC) method is used. We show that using the sampled realizations of the permeability field, the predictions can be significantly improved and the uncertainties can be assessed for this highly nonlinear problem."
}


%-------------------------2005-------------------------------
		

To improve the predictions in dynamic data driven simulations (DDDAS) for subsurface problems, we propose the permeability update based on observed measurements. Based on measurement errors and a priori information about the permeability field, such as covariance of permeability field and its values at the measurement locations, the permeability field is sampled. This sampling problem is highly nonlinear and Markov chain Monte Carlo (MCMC) method is used. We show that using the sampled realizations of the permeability field, the predictions can be significantly improved and the uncertainties can be assessed for this highly nonlinear problem.

@article {MR2150166,
    AUTHOR = {Chatzipantelidis, P. and Ginting, V. and Lazarov, R.},
     TITLE = {A {F}inite {V}olume {E}lement {M}ethod for a {N}on-{L}inear {E}lliptic
              {P}roblem},
  JOURNAL = {Numerical Linear Algebra with Applications},
    VOLUME = {12},
      YEAR = {2005},
    NUMBER = {5-6},
     PAGES = {515-546},
      ISSN = {1070-5325},
     CODEN = {NLAAEM},
   MRCLASS = {65N30},
  MRNUMBER = {2150166 (2006f:65115)},
MRREVIEWER = {Tom{\'a}{\v{s}} Roub{\'{\i}}{\v{c}}ek},
       DOI = {10.1002/nla.439},
       URL = {http://dx.doi.org/10.1002/nla.439},
 ABSTRACT="
 We consider a finite volume discretization of second-order non-linear elliptic boundary value problems on polygonal domains. Using relatively standard assumptions we show the existence of the finite volume solution. Furthermore, for a sufficiently small data the uniqueness of the finite volume solution may also be deduced. We derive error estimates in <i>H</i><sup>1</sup>-, <i>L</i><sub>2</sub>- and <i>L</i><sub>\infinity</sub>-norm for small data and convergence in <i>H</i><sup>1</sup>-norm for large data. In addition a Newton's method is analysed for the approximation of the finite volume solution and numerical experiments are presented.
 "
}

We consider a finite volume discretization of second-order non-linear elliptic boundary value problems on polygonal domains. Using relatively standard assumptions we show the existence of the finite volume solution. Furthermore, for a sufficiently small data the uniqueness of the finite volume solution may also be deduced. We derive error estimates in H¹-, L₂- and L_∈finity-norm for small data and convergence in H¹-norm for large data. In addition a Newton's method is analysed for the approximation of the finite volume solution and numerical experiments are presented.

@article {WRCR:WRCR10215,
author = {Efendiev, Y. and Datta-Gupta, A. and Ginting, V. and Ma, X. and Mallick, B.},
title = {An {E}fficient {T}wo-{S}tage Markov {C}hain Monte Carlo {M}ethod for {D}ynamic {D}ata {I}ntegration},
journal = {Water Resources Research},
volume = {41},
number = {12},
issn = {1944-7973},
url = {http://dx.doi.org/10.1029/2004WR003764},
doi = {10.1029/2004WR003764},
pages = {n/a--n/a},
keywords = {Uncertainty assessment, Uncertainty quantification, Inverse theory, MCMC, upscaling, acceptance rate},
year = {2005},
note = {W12423},
abstract="In this paper, we use a two-stage Markov chain Monte Carlo (MCMC) method for subsurface characterization that employs coarse-scale models. The purpose of the proposed method is to increase the acceptance rate of MCMC by using inexpensive coarse-scale runs based on single-phase upscaling. Numerical results demonstrate that our approach leads to a severalfold increase in the acceptance rate and provides a practical approach to uncertainty quantification during subsurface characterization."
}

%-------------------------2004-------------------------------
				

In this paper, we use a two-stage Markov chain Monte Carlo (MCMC) method for subsurface characterization that employs coarse-scale models. The purpose of the proposed method is to increase the acceptance rate of MCMC by using inexpensive coarse-scale runs based on single-phase upscaling. Numerical results demonstrate that our approach leads to a severalfold increase in the acceptance rate and provides a practical approach to uncertainty quantification during subsurface characterization.

@article {MR2146982,
    AUTHOR = {Ginting, V. and Ewing, R. and Efendiev, Y. and Lazarov, R.},
     TITLE = {Upscaled {M}odeling in {M}ultiphase {F}low {A}pplications},
  JOURNAL = {Computational \& Applied Mathematics},
    VOLUME = {23},
      YEAR = {2004},
    NUMBER = {2-3},
     PAGES = {213-233},
      ISSN = {1807-0302},
   MRCLASS = {76S05 (76M25 76T99)},
  MRNUMBER = {2146982 (2005m:76167)},
       DOI = {10.1590/S0101-82052004000200007},
       URL = {http://dx.doi.org/10.1590/S0101-82052004000200007},
  ABSTRACT="In this paper we consider upscaling of multiphase flow in porous media. We propose numerical techniques for upscaling of pressure and saturation equations. Extensions and applications of these approaches are considered in this paper. Numerical examples are presented."
}
		

In this paper we consider upscaling of multiphase flow in porous media. We propose numerical techniques for upscaling of pressure and saturation equations. Extensions and applications of these approaches are considered in this paper. Numerical examples are presented.

@article {MR2119929,
    AUTHOR = {Efendiev, Y. and Hou, T. and Ginting, V.},
     TITLE = {Multiscale {F}inite {E}lement {M}ethods for {N}onlinear {P}roblems and
              {T}heir {A}pplications},
  JOURNAL = {Communications in Mathematical Sciences},
    VOLUME = {2},
      YEAR = {2004},
    NUMBER = {4},
     PAGES = {553-589},
      ISSN = {1539-6746},
   MRCLASS = {65N30},
  MRNUMBER = {2119929 (2005m:65265)},
MRREVIEWER = {Karsten Urban},
       URL = {http://projecteuclid.org/euclid.cms/1109885498},
ABSTRACT="In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities."
}

In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities.

@article{MR2062582,
    AUTHOR = {Ginting, V.},
     TITLE = {Analysis of {T}wo-{S}cale {F}inite {V}olume {E}lement {M}ethod for
              {E}lliptic {P}roblem},
  JOURNAL = {Journal of Numerical Mathematics},
    VOLUME = {12},
      YEAR = {2004},
    NUMBER = {2},
     PAGES = {119-141},
      ISSN = {1570-2820},
   MRCLASS = {65N06 (65N30 76M12 76S05)},
  MRNUMBER = {2062582 (2005e:65165)},
MRREVIEWER = {Beny Neta},
       DOI = {10.1163/156939504323074513},
       URL = {http://dx.doi.org/10.1163/156939504323074513},
ABSTRACT="In this paper we propose and analyze a class of finite volume element method for solving a second order elliptic boundary value problem whose solution is defined in more than one length scales. The method has the ability to incorporate the small scale behaviors of the solution on the large scale one. This is achieved through the construction of the basis functions on each element that satisfy the homogeneous elliptic differential equation. Furthermore, the method enjoys numerical conservation feature which is highly desirable in many applications. Existing analyses on its finite element counterpart reveal that there exists a resonance error between the mesh size and the small length scale. This result motivates an oversampling technique to overcome this drawback. We develop an analysis of the proposed method under the assumption that the coefficients are of two scales and periodic in the small scale. The theoretical results are confirmed experimentally by several convergence tests. Moreover, we present an application of the method to flows in porous media."
}

In this paper we propose and analyze a class of finite volume element method for solving a second order elliptic boundary value problem whose solution is defined in more than one length scales. The method has the ability to incorporate the small scale behaviors of the solution on the large scale one. This is achieved through the construction of the basis functions on each element that satisfy the homogeneous elliptic differential equation. Furthermore, the method enjoys numerical conservation feature which is highly desirable in many applications. Existing analyses on its finite element counterpart reveal that there exists a resonance error between the mesh size and the small length scale. This result motivates an oversampling technique to overcome this drawback. We develop an analysis of the proposed method under the assumption that the coefficients are of two scales and periodic in the small scale. The theoretical results are confirmed experimentally by several convergence tests. Moreover, we present an application of the method to flows in porous media.