Discrete Maximum Principle for the Weak Galerkin Method for Anisotropic Diffusion Problems (Communications in Computational Physics, 2018)
A weak Galerkin discretization of the boundary value problem of a generalanisotropic diffusion problem is studied for preservation of the maximum principle. Itis shown that the direct application of the M-matrix theory to the stiffness matrix of theweak Galerkin discretization leads to a strong mesh condition requiring all of the meshdihedral angles to be strictly acute (a constant-order away from 90 degrees). To avoidthis difficulty, a reduced system is considered and shown to satisfy the discrete maximumprinciple under weaker mesh conditions. The discrete maximum principle isthen established for the full weak Galerkin approximation using the relations betweenthe degrees of freedom located on elements and edges. Sufficient mesh conditionsfor both piecewise constant and general anisotropic diffusion matrices are obtained.These conditions provide a guideline for practical mesh generation for preservation ofthe maximum principle. Numerical examples are presented.