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

Abstract

his 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 $(_Pk(T ), P_{k−1}(e), [P_{k+1}(T)]^d)$ elements with dimensions of space $d = 2, 3$ are employed and the numerical examples are tested to confirm the theory.