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

Abstract

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.

Funding

Nebraska Space Grant Consortium(80NSSC20M0112)