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.

中文翻译：

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非均匀矩形分区上弱伽辽金有限元法的离散极大值原理

本文建立了离散最大原理（DMP），用于近似求解非均匀矩形分区上的弱伽辽金有限元方法得到的对流-扩散-反应问题。DMP 分析基于弱伽辽金的简化公式，仅涉及定义在每个单元边界上的近似函数。简化的弱伽辽金方法比通常的弱伽辽金方法具有降低的计算复杂度，并且确实在反应项出现时提供了不同于弱伽辽金的离散化方案。简化的弱伽辽金在均匀矩形分区上的应用为二阶椭圆方程产生了一些 $5$-和 $7$-点有限差分格式。