A Weak Galerkin Finite Element Method for the Stokes Equations

TL;DR

This paper presents a novel weak Galerkin finite element method for solving the Stokes equations, allowing for arbitrary polygonal or polyhedral meshes and providing optimal error estimates.

Contribution

It introduces a stable, discontinuous weak Galerkin finite element scheme for the Stokes equations on arbitrary shape meshes with proven optimal error bounds.

Findings

Achieved optimal-order error estimates in various norms.

Applicable to meshes with arbitrary polygonal or polyhedral shapes.

Demonstrated stability and convergence of the method.

Abstract

This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primary velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree $k\ge 1$ for the velocity and polynomials of degree $k-1$ for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree $k-1$ on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.