Construction of Locally Conservative Fluxes for High Order Continuous Galerkin Finite Element Methods. Deng, Q., Ginting, V., & McCaskill, B. Journal of Computational and Applied Mathematics, 359:166-181, 2019.
Paper doi abstract bibtex 5 downloads 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{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."
}
Downloads: 5
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