A Weak Galerkin Finite Element Scheme for solving the stationary Stokes Equations

TL;DR

This paper introduces a novel weak Galerkin finite element method for stationary Stokes equations, utilizing discontinuous polynomials and a unique gradient operator, with proven optimal error estimates and supporting numerical results.

Contribution

The paper develops a new WG finite element scheme based on a different gradient operator, offering flexibility with discontinuous functions on arbitrary polygons or polyhedra.

Findings

Optimal error estimates in various norms

Numerical results confirm theoretical accuracy

Flexible use of discontinuous functions

Abstract

A weak Galerkin (WG) finite element method for solving the stationary Stokes equations in two- or three- dimensional spaces by using discontinuous piecewise polynomials is developed and analyzed. The variational form we considered is based on two gradient operators which is different from the usual gradient-divergence operators. The WG method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order error estimates are established for the corresponding WG finite element solutions in various norms. Numerical results are presented to illustrate the theoretical analysis of the new WG finite element scheme for Stokes problems.