A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

Abstract

In this article, a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed. The idea of using the $P_k$-harmonic polynomial space instead of the full polynomial space $P_{k}$ is to use a much smaller number of basis functions to achieve the same accuracy when $k\geqslant 2$. The optimal rate of convergence is derived in both $H^1$ and $L^2$ norms. Numerical experiments have been conducted to verify the theoretical error estimates. In addition, numerical comparisons of using the $P_{2}$-harmonic polynomial space and using the standard $P_2$ polynomial space are presented.