The lowest-degree stabilizer-free weak Galerkin finite element method for Poisson equation on rectangular and triangular meshes

Recently, the study on weak Galerkin (WG) methods with or without stabilizer parameters have received much attention. The WG methods are a discontinuous extension of the standard finite element methods in which classical differential operators are approximated on functions with discontinuity. A stabilizer term in the WG formulation is used to guarantee convergence and stability of the discontinuous approximations for a model problem. By removing this parameter, we can reduce the complexity of programming on this numerical method. Our goal in this paper is to introduce a new stabilizer-free WG (SFWG) method to solve the Poisson equation in which we use a new combination of WG elements. Numerical experiments indicate that our SFWG scheme is faster and more economical than the standard WG scheme. Errors and convergence rates on two types of mesh are presented for each of the considered methods, which show that our numerical scheme has O(h(2)) convergence rate while another method has 0(h) convergence rate in the energy norm and the L-2-norm.