A velocity-based moving mesh Discontinuous Galerkin method for the advection-diffusion equation

In convection-dominated flows, robustness of the spatial discretisation is a key property. While Interior Penalty Galerkin (IPG) methods already proved efficient in the situation of large mesh Peclet numbers, Arbitrary Lagrangian-Eulerian (ALE) methods are able to reduce the convection-dominance by moving the mesh. In this paper, we introduce and analyse a velocity-based moving mesh discontinuous Galerkin (DG) method for the solution of the linear advection-diffusion equation. By introducing a smooth parameterized velocity $\Tilde{V}$ that separates the flow into a mean flow, also called moving mesh velocity, and a remaining advection field $V-\Tilde{V}$, 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 time-stepping. Finally, by adapting the existing robust error criteria to this moving mesh situation, we derived robust \textit{a posteriori} error criteria that describe the potentially small deviation to the mean flow and include the information of a transition towards $V=\Tilde{V}$.