Volume 18, Issue 1
Discrete Maximum Principle for the Weak Galerkin Method for Anisotropic Diffusion Problems

Weizhang Huang & Yanqiu Wang

Commun. Comput. Phys., 18 (2015), pp. 65-90.

Published online: 2018-04

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  • Abstract

A weak Galerkin discretization of the boundary value problem of a general anisotropic diffusion problem is studied for preservation of the maximum principle. It is shown that the direct application of the M-matrix theory to the stiffness matrix of the weak Galerkin discretization leads to a strong mesh condition requiring all of the mesh dihedral angles to be strictly acute (a constant-order away from 90 degrees). To avoid this difficulty, a reduced system is considered and shown to satisfy the discrete maximum principle under weaker mesh conditions. The discrete maximum principle is then established for the full weak Galerkin approximation using the relations between the degrees of freedom located on elements and edges. Sufficient mesh conditions for both piecewise constant and general anisotropic diffusion matrices are obtained. These conditions provide a guideline for practical mesh generation for preservation of the maximum principle. Numerical examples are presented.

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@Article{CiCP-18-65, author = {}, title = {Discrete Maximum Principle for the Weak Galerkin Method for Anisotropic Diffusion Problems}, journal = {Communications in Computational Physics}, year = {2018}, volume = {18}, number = {1}, pages = {65--90}, abstract = {

A weak Galerkin discretization of the boundary value problem of a general anisotropic diffusion problem is studied for preservation of the maximum principle. It is shown that the direct application of the M-matrix theory to the stiffness matrix of the weak Galerkin discretization leads to a strong mesh condition requiring all of the mesh dihedral angles to be strictly acute (a constant-order away from 90 degrees). To avoid this difficulty, a reduced system is considered and shown to satisfy the discrete maximum principle under weaker mesh conditions. The discrete maximum principle is then established for the full weak Galerkin approximation using the relations between the degrees of freedom located on elements and edges. Sufficient mesh conditions for both piecewise constant and general anisotropic diffusion matrices are obtained. These conditions provide a guideline for practical mesh generation for preservation of the maximum principle. Numerical examples are presented.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.180914.121214a}, url = {http://global-sci.org/intro/article_detail/cicp/11018.html} }
TY - JOUR T1 - Discrete Maximum Principle for the Weak Galerkin Method for Anisotropic Diffusion Problems JO - Communications in Computational Physics VL - 1 SP - 65 EP - 90 PY - 2018 DA - 2018/04 SN - 18 DO - http://doi.org/10.4208/cicp.180914.121214a UR - https://global-sci.org/intro/article_detail/cicp/11018.html KW - AB -

A weak Galerkin discretization of the boundary value problem of a general anisotropic diffusion problem is studied for preservation of the maximum principle. It is shown that the direct application of the M-matrix theory to the stiffness matrix of the weak Galerkin discretization leads to a strong mesh condition requiring all of the mesh dihedral angles to be strictly acute (a constant-order away from 90 degrees). To avoid this difficulty, a reduced system is considered and shown to satisfy the discrete maximum principle under weaker mesh conditions. The discrete maximum principle is then established for the full weak Galerkin approximation using the relations between the degrees of freedom located on elements and edges. Sufficient mesh conditions for both piecewise constant and general anisotropic diffusion matrices are obtained. These conditions provide a guideline for practical mesh generation for preservation of the maximum principle. Numerical examples are presented.

Weizhang Huang & Yanqiu Wang. (2020). Discrete Maximum Principle for the Weak Galerkin Method for Anisotropic Diffusion Problems. Communications in Computational Physics. 18 (1). 65-90. doi:10.4208/cicp.180914.121214a
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