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

TL;DR

This paper proves a discrete maximum principle for a simplified weak Galerkin finite element method applied to convection-diffusion-reaction problems on nonuniform rectangular grids, offering reduced computational complexity and new finite difference schemes.

Contribution

It introduces a simplified weak Galerkin method that maintains the discrete maximum principle and reduces computational complexity, differing from traditional weak Galerkin approaches.

Findings

Established a discrete maximum principle for the simplified weak Galerkin method.

Derived new finite difference schemes from the simplified method.

Numerical experiments confirmed the scheme's accuracy and maximum principle.

Abstract

This article establishes a discrete maximum principle (DMP) for the approximate solution of convection-diffusion-reaction problems obtained from the weak Galerkin finite element method on nonuniform rectangular partitions. The DMP analysis is based on a simplified formulation of the weak Galerkin involving only the approximating functions defined on the boundary of each element. The simplified weak Galerkin method has a reduced computational complexity over the usual weak Galerkin, and indeed provides a discretization scheme different from the weak Galerkin when the reaction term presents. An application of the simplified weak Galerkin on uniform rectangular partitions yields some $5$- and $7$-point finite difference schemes for the second order elliptic equation. Numerical experiments are presented to verify the discrete maximum principle and the accuracy of the scheme, particularly the finite difference scheme.