Adaptive Discontinuous Galerkin Methods for 1D unsteady Convection-Diffusion Problems on a Moving Mesh

Abstract

In convection-dominated flows, large scale trends necessarily coexist with small-scale effects. While reducing the convection-dominance by moving the mesh, also called Arbitrary Lagrangian-Eulerian (ALE), already proved efficient, Adaptive Mesh Refinement (AMR) is able to catch the small scale effects. But, ALE introduces uncertainties that cannot be neglected in front of the small scale effects, therefore it is unsatisfying to use AMR the same way in an ALE situation as we do on static meshes. In this paper, an h-refinement criterion is built up and tested by studying the approximation error of a moving mesh, interior penalty discontinuous Galerkin (DG) semi-discretization of the 1D nonstationary convection-diffusion equation. It is done through three hotspots: the uncertainties due to the mesh movement, the error sources and the error propagation. Whereas the uncertainties as well as the error sources are easily measurable, having a precise understanding of the error propagation remains difficult. The cheap and efficient way to have a faithful picture of the error propagation in a dynamic situation, is by measuring the residual of the approximation solution. In addition to a stabilizing effect of the moving mesh, this method provides interesting results: while ALE approximates in terms of the L²-norm, the developed refinement criterion spots early where the H⁻¹-norm of the approximation error will explode.