Weak Galerkin finite element methods with and without stabilizers for H(div;Ω)${\bf H}(\mbox{div}; \Omega )$‐elliptic problems

Abstract

In this article, we propose the weak Galerkin (WG) finite element schemes for H(div;Ω)${\bf H}(\mbox{div}; {\Omega })$ ‐elliptic problems with and without stabilizers. Optimal orders of convergence are established for the WG approximations in both discrete energy norm and L2 norm. Removing stabilizers from WG finite element methods will simplify the formulations, reduce programming complexity, and may also speed up the computation time. More precisely, for sufficiently smooth solutions, we have proved the supercloseness of order two for the stabilizer free weak Galerkin finite element solution. Several numerical tests are presented to demonstrate the effectiveness of our method.