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

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), Pk-1(e), [Pk+1(T)](d)) elements with dimensions of space d = 2, 3 are employed and the numerical examples are tested to confirm the theory.