A Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear Defect Correction Strategy for the Numerical Solution of Steady State Diffusion Problems in Heterogeneous and Anisotropic Media Satisfying the Discrete Maximum Principle

In the present paper, we solve the steady state diffusion equation in 3D domains by means of a cell-centered finite volume method that uses a Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear defect correction strategy (MPFA-DNL) to guarantee the Discrete Maximum Principle (DMP)....

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Veröffentlicht in:	Journal of scientific computing Jg. 93; H. 2; S. 42
Hauptverfasser:	Cavalcante, T. M., Filho, R. J. M. Lira, Souza, A. C. R., Carvalho, D. K. E., Lyra, P. R. M.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.11.2022 Springer Nature B.V
Schlagworte:	Algorithms Anisotropic media Anisotropy Approximation Computational Mathematics and Numerical Analysis Defects Diamonds Diffusion rate Finite element analysis Finite volume method Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Maximum principle Partial differential equations Robustness (mathematics) Steady state Tensors Theoretical Heterogeneous and anisotropic media 3D diffusion problems MPFA-DNL Discrete maximum principle (DMP) Unstructured tetrahedral meshes
ISSN:	0885-7474, 1573-7691
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Abstract	In the present paper, we solve the steady state diffusion equation in 3D domains by means of a cell-centered finite volume method that uses a Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear defect correction strategy (MPFA-DNL) to guarantee the Discrete Maximum Principle (DMP). Our formulation is based in the fact that the flux of MPFA methods can be split into two different parts: a Two Point Flux Approximation (TPFA) component and the Cross-Diffusion Terms (CDT). In the linear MPFA-D method, this split is particularly simple since it lies at the core of the original method construction. In this context, we introduce a non-linear defect correction, aiming to mitigate, whenever necessary, the contributions from the CDT, avoiding, this way, spurious oscillations and DMP violations. Our new MPFA-DNL scheme is locally conservative and capable of dealing with arbitrary anisotropic diffusion tensors and unstructured meshes, without harming the second order convergence rates of the original MPFA-D. To appraise the accuracy and robustness of our formulation, we solve some benchmark problems found in literature. In this paper, we restrict ourselves to tetrahedral meshes, even though, in principle, there is no restriction to extend the method to other polyhedral control volumes.
AbstractList	In the present paper, we solve the steady state diffusion equation in 3D domains by means of a cell-centered finite volume method that uses a Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear defect correction strategy (MPFA-DNL) to guarantee the Discrete Maximum Principle (DMP). Our formulation is based in the fact that the flux of MPFA methods can be split into two different parts: a Two Point Flux Approximation (TPFA) component and the Cross-Diffusion Terms (CDT). In the linear MPFA-D method, this split is particularly simple since it lies at the core of the original method construction. In this context, we introduce a non-linear defect correction, aiming to mitigate, whenever necessary, the contributions from the CDT, avoiding, this way, spurious oscillations and DMP violations. Our new MPFA-DNL scheme is locally conservative and capable of dealing with arbitrary anisotropic diffusion tensors and unstructured meshes, without harming the second order convergence rates of the original MPFA-D. To appraise the accuracy and robustness of our formulation, we solve some benchmark problems found in literature. In this paper, we restrict ourselves to tetrahedral meshes, even though, in principle, there is no restriction to extend the method to other polyhedral control volumes.
ArticleNumber	42
Author	Souza, A. C. R. Cavalcante, T. M. Filho, R. J. M. Lira Lyra, P. R. M. Carvalho, D. K. E.
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Cites_doi	10.1002/fld.3850 10.1007/978-3-642-85949-6 10.1007/s10596-010-9191-5 10.1002/fld.1948 10.1002/fld.2496 10.1007/s00211-013-0545-5 10.1016/j.compstruc.2021.106510 10.1016/j.jcp.2009.01.031 10.1016/j.crma.2004.02.010 10.1090/S0025-5718-00-01270-9 10.1137/16M1098000 10.1016/j.jcp.2018.06.052 10.1016/j.compfluid.2015.11.013 10.1006/jcph.2000.6418 10.1016/j.jcp.2014.07.003 10.1016/j.apnum.2020.08.008 10.1002/num.20320 10.1002/zamm.201900320 10.1515/RJNAMM.2009.014 10.1016/j.apnum.2020.04.014 10.1016/j.jcp.2007.09.021 10.1002/fld.2258 10.1137/S1064827595293594 10.1016/0045-7825(73)90019-4 10.1002/fld.4829 10.1137/S1064827595293582 10.1023/A:1011510505406 10.1007/978-3-642-59721-3_5 10.1007/978-3-642-20671-9_89 10.1016/S0924-6509(06)80018-8 10.1007/978-3-540-78319-0_1 10.1016/S1874-5733(04)80010-X 10.1007/BFb0064470 10.1137/1.9780898719208
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Snippet	In the present paper, we solve the steady state diffusion equation in 3D domains by means of a cell-centered finite volume method that uses a Multipoint Flux...
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SubjectTerms	Algorithms Anisotropic media Anisotropy Approximation Computational Mathematics and Numerical Analysis Defects Diamonds Diffusion rate Finite element analysis Finite volume method Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Maximum principle Partial differential equations Robustness (mathematics) Steady state Tensors Theoretical
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Title	A Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear Defect Correction Strategy for the Numerical Solution of Steady State Diffusion Problems in Heterogeneous and Anisotropic Media Satisfying the Discrete Maximum Principle
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