Superconvergence of Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

Abstract

In this paper, we present a unified approach to study superconvergence behavior of the local discontinuous Galerkin (LDG) method for high-order time-dependent partial differential equations. We select the third and fourth order equations as our models to demonstrate this approach and the main idea. Superconvergence results for the solution itself and the auxiliary variables are established. To be more precise, we first prove that, for any polynomial of degree k, the errors of numerical fluxes at nodes and for the cell averages are superconvergent under some suitable initial discretization, with an order of $$O(h^{2k+1})$$ . We then prove that the LDG solution is $$(k+2)$$ -th order superconvergent towards a particular projection of the exact solution and the auxiliary variables. As byproducts, we obtain a $$(k+1)$$ -th and $$(k+2)$$ -th order superconvergence rate for the derivative and function value approximation separately at a class of Radau points. Moreover, for the auxiliary variables, we, for the first time, prove that the convergence rate of the derivative error at the interior Radau points can reach as high as $$k+2$$ . Numerical experiments demonstrate that most of our error estimates are optimal, i.e., the error bounds are sharp.