Element-by-Element Post-Processing of Discontinuous Galerkin Methods for Naghdi Arches

Abstract

In this paper, we consider discontinuous Galerkin approximations to the solution of Naghdi arches and show how to post-process them in an element-by-element fashion to obtain a far better approximation. Indeed, we prove that, if polynomials of degree $k$ are used, the post-processed approximation converges with order $2k+1$ in the $L^2$-norm throughout the domain. This has to be contrasted with the fact that before post-processing, the approximation converges with order $k + 1$ only. Moreover, we show that this superconvergence property does not deteriorate as the thickness of the arch becomes extremely small. Numerical experiments verifying the above-mentioned theoretical results are displayed.