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

Discrete maximum principle for the weak Galerkin method for anisotropic diffusion problems