A Discrete Divergence-Free Weak Galerkin Finite Element Method for the Stokes Equations

TL;DR

This paper introduces a novel divergence-free weak Galerkin finite element method for the Stokes equations that simplifies the computational system by eliminating pressure variables, applicable to complex meshes.

Contribution

The paper develops explicit divergence-free bases for weak Galerkin elements on arbitrary meshes, enabling pressure elimination and reducing system complexity.

Findings

Reduces saddle point problem to symmetric positive definite system

Demonstrates robustness and accuracy through numerical tests

Applicable to general polygonal and polyhedral meshes

Abstract

A discrete divergence-free weak Galerkin finite element method is developed for the Stokes equations based on a weak Galerkin (WG) method introduced in the reference [15]. Discrete divergence-free bases are constructed explicitly for the lowest order weak Galerkin elements in two and three dimensional spaces. These basis functions can be derived on general meshes of arbitrary shape of polygons and polyhedrons. With the divergence-free basis derived, the discrete divergence-free WG scheme can eliminate the pressure variable from the system and reduces a saddle point problem to a symmetric and positive definite system with many fewer unknowns. Numerical results are presented to demonstrate the robustness and accuracy of this discrete divergence-free WG method.