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

In this paper, we develop a novel discontinuous Galerkin (DG) finite element method for solving the Poisson's equation u x x + u y y = 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 L 2 -norm of O (h p + 1) + O (k q + 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. [ABSTRACT FROM AUTHOR]