On Multiscale Methods in Petrov-Galerkin Formulation

On Multiscale Methods in Petrov-Galerkin Formulation. Elfverson, D., Ginting, V., & Henning, P. Numerische Mathematik, 131(4):643-682, Springer Berlin Heidelberg, 2015.

Paper doi abstract bibtex

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{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."
}

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