Volume 26, Issue 1
A Nonlinear Finite Volume Element Method Satisfying Maximum Principle for Anisotropic Diffusion Problems on Arbitrary Triangular Meshes

Yanni Gao, Shuai Wang, Guangwei Yuan & Xudeng Hang

Commun. Comput. Phys., 26 (2019), pp. 135-159.

Published online: 2019-02

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  • Abstract

A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed. Starting with a standard linear conforming finite volume element approximation, a corrective term with respect to the flux jumps across element boundaries is added to make the scheme satisfy the discrete maximum principle. The new scheme is free of the anisotropic non-obtuse angle condition which is a severe restriction on the grids for problems with anisotropic diffusion. Moreover, this manipulation can nearly keep the same accuracy as the original scheme. We prove the existence of the numerical solution for this nonlinear scheme theoretically. Numerical results and a grid convergence study are presented for both continuous and discontinuous anisotropic diffusion problems.

  • Keywords

Finite volume element method, nonlinear correction, discrete maximum principle, anisotropic diffusion.

  • AMS Subject Headings

65N08, 65N12, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-26-135, author = {}, title = {A Nonlinear Finite Volume Element Method Satisfying Maximum Principle for Anisotropic Diffusion Problems on Arbitrary Triangular Meshes}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {1}, pages = {135--159}, abstract = {

A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed. Starting with a standard linear conforming finite volume element approximation, a corrective term with respect to the flux jumps across element boundaries is added to make the scheme satisfy the discrete maximum principle. The new scheme is free of the anisotropic non-obtuse angle condition which is a severe restriction on the grids for problems with anisotropic diffusion. Moreover, this manipulation can nearly keep the same accuracy as the original scheme. We prove the existence of the numerical solution for this nonlinear scheme theoretically. Numerical results and a grid convergence study are presented for both continuous and discontinuous anisotropic diffusion problems.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0261}, url = {http://global-sci.org/intro/article_detail/cicp/13029.html} }
TY - JOUR T1 - A Nonlinear Finite Volume Element Method Satisfying Maximum Principle for Anisotropic Diffusion Problems on Arbitrary Triangular Meshes JO - Communications in Computational Physics VL - 1 SP - 135 EP - 159 PY - 2019 DA - 2019/02 SN - 26 DO - http://dor.org/10.4208/cicp.OA-2017-0261 UR - https://global-sci.org/intro/article_detail/cicp/13029.html KW - Finite volume element method, nonlinear correction, discrete maximum principle, anisotropic diffusion. AB -

A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed. Starting with a standard linear conforming finite volume element approximation, a corrective term with respect to the flux jumps across element boundaries is added to make the scheme satisfy the discrete maximum principle. The new scheme is free of the anisotropic non-obtuse angle condition which is a severe restriction on the grids for problems with anisotropic diffusion. Moreover, this manipulation can nearly keep the same accuracy as the original scheme. We prove the existence of the numerical solution for this nonlinear scheme theoretically. Numerical results and a grid convergence study are presented for both continuous and discontinuous anisotropic diffusion problems.

Yanni Gao, Shuai Wang, Guangwei Yuan & Xudeng Hang. (2019). A Nonlinear Finite Volume Element Method Satisfying Maximum Principle for Anisotropic Diffusion Problems on Arbitrary Triangular Meshes. Communications in Computational Physics. 26 (1). 135-159. doi:10.4208/cicp.OA-2017-0261
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