A symmetric weak Galerkin method for solving non-divergence form elliptic equations

Abstract

We reformulate the second order non-divergence form elliptic equations in a symmetric variation form, i.e., a least-squares form. We design a weak Galerkin finite element method for this high-regularity formulation. The optimal order of convergence is proved. Numerical results verify the theory. In addition, numerical results, compared to the existing weak Galerkin method for the non-symmetric form, and the new methods show advantage on the accuracy and the degree of freedoms.