A stabilizer free weak Galerkin finite element method for elliptic equation with lower regularity

摘要: This paper presents error analysis of stabilizer free weak Galerkin finite element method (SFWG-FEM) for a second order elliptic equation 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.