Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations

In this paper, we present the negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method solving nonlinear hyperbolic equations with smooth solutions. The smoothness-increasing accuracy-conserving (SIAC) filter is a post-processing technique to enhance the accuracy of the discontinuous Galerkin (DG) solutions. This work is the essential step to extend the SIAC filter to the moving mesh for nonlinear problems. By the post-processing theory, the negative norm estimates are vital to get the superconvergence error estimates of the solutions after post-processing in the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} norm. Although the SIAC filter has been extended to nonuniform mesh, the analysis of filtered solutions on the nonuniform mesh is complicated. We prove superconvergence error estimates in the negative norm for the ALE-DG method on moving meshes. The main difficulties of the analysis are the terms in the ALE-DG scheme brought by the grid velocity field, and the time-dependent function space. The mapping from time-dependent cells to reference cells is very crucial in the proof. The numerical results also confirm the theoretical proof.