Development of a LDG method on polytopal mesh with optimal order of convergence

Abstract

A Local discontinuous Galerkin (LDG) finite element method on triangular mesh was introduced and analyzed in Castillo et al. (2000) for elliptic problem with suboptimal order of convergence for the flux variable. The purpose of this work is to develop a LDG finite element method on polytopal mesh that achieves optimal convergence rate for both unknowns, potential u and its gradient q. In our new LDG method, u and q are approximated by polynomials of degree k and k−1 respectively. Optimal order of convergence are obtained for both unknowns. Numerical results in 2d and 3d are presented to confirm the theory.