A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids

Abstract

. In this paper, we design and analyze new residual-type a posteriori error estimators for the local discontinuous Galerkin (LDG) method applied to semilinear second-order elliptic problems in two dimensions of the type (cid:0) ∆ u = f ( x ;u ). We use our recent superconvergence results derived in Commun. Appl. Math. Comput. (2021) to prove that the LDG solution is superconvergent with an order p +2 towards the p -degree right Radau interpolating polynomial of the exact solution, when tensor product polynomials of degree at most p are considered as basis for the LDG method. Moreover, we show that the global discretization error can be decomposed into the sum of two errors. The first error can be expressed as a linear combination of two ( p +1)- degree Radau polynomials in the x - and y (cid:0) directions. The second error converges to zero with order p + 2 in the L 2 -norm. This new result allows us to construct a posteriori error estimators of residual type. We prove that the proposed a posteriori error estimators converge to the true errors in the L 2 -norm under mesh refinement at the optimal rate. The order of convergence is proved to be p + 2. We further prove that our a posteriori error estimates yield upper and lower bounds for the actual error. Finally, a series of numerical examples are presented to validate the theoretical results and numerically demonstrate the convergence of the proposed a posteriori error estimators.