Error analysis of a novel discontinuous Galerkin method for the two-dimensional Poisson's equation

Abstract

In this paper, we develop a novel discontinuous Galerkin (DG) finite element method for solving the Poisson's equation uxx+uyy=f(x,y) on Cartesian grids. The proposed method consists of first applying the standard DG method in the x-spatial variable leading to a system of ordinary differential equations (ODEs) in the y-variable. Then, using the method of line, the DG method is directly applied to discretize the resulting system of ODEs. In fact, we propose a fully DG scheme that uses p-th and q-th degree DG methods in the x and y variables, respectively. We show that, under proper choices of numerical fluxes, the method achieves optimal convergence rate in the L2-norm of O(hp+1)+O(kq+1) for the DG solution, where h and k denote, respectively, the mesh step sizes for the x and y variables. Our theoretical results are validated through several numerical experiments.