Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second-Order Elliptic Problems on Cartesian Grids

This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin (LDG) method for two-dimensional semilinear second-order elliptic problems of the form -Delta u = f(x, y, u) on Cartesian grids. By introducing special Gauss-Radau projections and using duality arguments, we obtain, under some suitable choice of numerical fluxes, the optimal convergence order in L-2-norm of O(h(p+1)) for the LDG solution and its gradient, when tensor product polynomials of degree at most p and grid size h are used. Moreover, we prove that the LDG solutions are superconvergent with an order p + 2 toward particular Gauss-Radau projections of the exact solutions. Finally, we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves (p + 1)-th order superconvergence. Some numerical experiments are performed to illustrate the theoretical results.