A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

Abstract

This paper presents error analysis of a stabilizer free weak Galerkin finite element method (SFWG-FEM) for second-order elliptic equations with low regularity solutions. The standard error analysis of SFWG-FEM requires additional regularity on solutions, such as H 2 -regularity for the second-order convergence. However, if the solutions are in H 1+s with 0 < s < 1, numerical experiments show that the SFWG-FEM is also effective and stable with the (1 + s)-order convergence rate, so we develop a theoretical analysis for it. We introduce a standard H 2 finite element approximation for the elliptic problem, and then we apply the SFWG-FEM to approach this smooth approximating finite element solution. Finally, we establish the error analysis for SFWG-FEM with low regularity in both discrete H 1 -norm and standard L 2 -norm. The (P k (T ), P k−1 (e), [P k+1 (T )] d ) elements with dimensions of space d = 2, 3 are employed and the numerical examples are tested to confirm the theory.