  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.
