Generalized Weak Galerkin Methods For Stokes Equations

TL;DR

This paper introduces a generalized weak Galerkin finite element method for solving Stokes equations, providing error estimates and numerical verification of its effectiveness and robustness.

Contribution

It develops a new weak Galerkin method with a novel weak gradient definition applicable to arbitrary polynomial combinations.

Findings

Error estimates in energy and L2 norms are established.

Numerical examples confirm the theoretical convergence and robustness.

The method is effective for various polynomial combinations.

Abstract

A new weak Galerkin finite element method, called generalized weak Galerkin method ({g}WG), is introduced for Stokes equations in this paper by using a new definition of the weak gradient. Error estimates in energy norm and $L^2$ norm for the velocity and $L^2$ norm for the pressure are derived for elements with arbitrary combination of polynomials. Some numerical examples are presented to verify the effectiveness, theoretical convergence orders, and robustness of the proposed scheme.