A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions
This work provides a theoretical guarantee (DMP) for a specific numerical method on nonuniform meshes, which is important for practitioners using weak Galerkin methods but is incremental in nature.
The authors prove a discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions for convection-diffusion-reaction problems, using a simplified formulation that reduces computational complexity. Numerical experiments verify the DMP and accuracy, including a new 5- and 7-point finite difference scheme on uniform partitions.
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.