A Velocity-Based Moving Mesh Discontinuous Galerkin Method for the Advection-Diffusion Equation

Abstract In convection-dominated flows, robustness of the spatial discretization is a key property. While interior penalty Galerkin (IPG) methods have already proved efficient in the situation of large mesh Peclet numbers, arbitrary Lagrangian-Eulerian (ALE) methods can reduce the convection-dominance by moving the mesh. In this paper, we introduce and analyze a velocity-based moving mesh discontinuous Galerkin (DG) method for the solution of the linear advection-diffusion equation. By introducing a smooth parameterized velocity $$\tilde{{\textbf {V}}}$$ V ~ that separates the flow into a mean flow, also called moving mesh velocity, and a remaining advection field $${\textbf {V}}-\tilde{{\textbf {V}}}$$ V - V ~ , we made a convergence analysis based on the smoothness of the mesh velocity. Furthermore, the reduction of the advection speed improves the stability of an explicit timestepping. Finally, by adapting the existing robust error criteria to this moving mesh situation, we derived robust a posteriori error criteria that describe the potentially small deviation from the mean flow and include the information of a transition towards $${\textbf {V}}=\tilde{{\textbf {V}}}$$ V = V ~ .