A primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form

Source abstract

This article proposes a new numerical algorithm for second order elliptic equations in non-divergence form. The new method is based on a discrete weak Hessian operator locally constructed by following the weak Galerkin strategy. The numerical solution is characterized as a minimization of a non-negative quadratic functional with constraints that mimic the second order elliptic equation by using the discrete weak Hessian. The resulting Euler-Lagrange equation offers a symmetric finite element scheme involving both the primal and a dual variable known as the Lagrange multiplier, and thus the name of primal-dual weak Galerkin finite element method . Error estimates of optimal order are derived for the corresponding finite element approximations in a discrete H 2 H^2 -norm, as well as the usual H 1 H^1 - and L 2 L^2 -norms. The convergence theory is based on the assumption that the solution of the model problem is H 2 H^2 -regular, and that the coefficient tensor in the PDE is piecewise continuous and uniformly positive definite in the domain. Some numerical results are presented for smooth and non-smooth coefficients on convex and non-convex domains, which not only confirm the developed convergence theory but also a superconvergence result.