Variational Multiscale Finite Element Method for Flows in Highly Porous Media

O. Iliev, R. Lazarov, J. Willems

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Abstract

We present a two-scale finite element method (FEM) for solving Brinkman's and Darcy's equations. These systems of equations model fluid flows in highly porous and porous media, respectively. The method uses a recently proposed discontinuous Galerkin FEM for Stokes' equations by Wang and Ye and the concept of subgrid approximation developed by Arbogast for Darcy's equations. In order to reduce the "resonance error" and to ensure convergence to the global fine solution, the algorithm is put in the framework of alternating Schwarz iterations using subdomains around the coarse-grid boundaries. The discussed algorithms are implemented using the Deal.II finite element library and are tested on a number of model problems. © 2011 Society for Industrial and Applied Mathematics.
Original languageEnglish (US)
Pages (from-to)1350-1372
Number of pages23
JournalMultiscale Modeling & Simulation
Volume9
Issue number4
DOIs
StatePublished - Oct 2011
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: Fraunhofer Institut fur Techno- und Wirtschaftsmathematik, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany, and Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str., bl. 8, 1113, Sofia, Bulgaria (oleg.iliev@itwm.fraunhofer.de). This author's research was supported by DFG project "Multiscale analysis of two-phase flow in porous media with complex heterogeneities."Department of Mathematics, Texas A&M University, College Station, TX 77843, and Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str., bl. 8, 1113, Sofia, Bulgaria (lazarov@math.tamu.edu). This author's research was supported in parts by award KUS-C1-016-04 made by King Abdullah University of Science and Technology (KAUST) and by NSF grant DMS-1016525.Department of Mathematics, Texas A&M University, College Station, TX 77843 (jwillems@math.tamu.edu). This author's research was supported by DAAD-PPP D/07/10578, NSF grants DMS-0713829 and DMS-1016525, and the Studienstiftung des deutschen Volkes (German National Academic Foundation).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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