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

Abstract

A stabilizer-free weak Galerkin finite element method is proposed for the Stokes equations in this paper. Here we omit the stabilizer term in the new method by increasing the degree of polynomial approximating spaces for the weak gradient operators. The new algorithm is simple in formulation and the computational complexity is also reduced. The corresponding approximating spaces consist of piecewise polynomials of degree $k\geq1$ for the velocity and $k-1$ for the pressure, respectively. Optimal order error estimates have been derived for the velocity in both $H^1$ and $L^2$ norms and for the pressure in $L^2$ norm. Numerical examples are presented to illustrate the accuracy and convergency of the method.