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This article establishes a discrete maximum principle (DMP) for theapproximate solution of convection-diffusion-reaction problems obtained fromthe weak Galerkin finite element method on nonuniform rectangular partitions.The DMP analysis is based on a simplified formulation of the weak Galerkininvolving only the approximating functions defined on the boundary of eachelement. The simplified weak Galerkin method has a reduced computationalcomplexity over the usual weak Galerkin, and indeed provides a discretizationscheme different from the weak Galerkin when the reaction term presents. Anapplication of the simplified weak Galerkin on uniform rectangular partitionsyields some $5$- and $7$-point finite difference schemes for the second orderelliptic equation. Numerical experiments are presented to verify the discretemaximum principle and the accuracy of the scheme, particularly the finitedifference scheme.

A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions