On stabilizer-free weak Galerkin finite element methods on polytopal meshes

A stabilizing/penalty term is often used in finite element methods with discontinuous approximations to enforce connection of discontinuous functions across element boundaries. Removing stabilizers from discontinuous finite element methods will simplify the formulations and reduce programming complexity significantly. The goal of this paper is to introduce a stabilizer free weak Galerkin finite element method for second order elliptic equations on polytopal meshes in 2D or 3D. This new WG method keeps a simple symmetric positive definite form and can work on polygonal/polyheral meshes. Optimal order error estimates are established for the corresponding WG approximations in both a discrete $H^1$ norm and the $L^2$ norm. Numerical results are presented verifying the theorem.