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 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 and the use of the nonconservative ALE formulation changes the coercivity condition. 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 $V=\tilde{V}$.

翻译：在对流主导流动中，空间离散化的鲁棒性是一个关键属性。虽然内部惩罚Galerkin方法在大网格佩克莱数情况下已被证明有效，但任意拉格朗日-欧拉方法能够通过移动网格来降低对流主导性。本文针对线性对流扩散方程的求解，提出并分析了一种基于速度的移动网格间断Galerkin方法。通过引入平滑参数化速度$\tilde{V}$，将流动分解为平均流（也称为移动网格速度）和剩余的对流场$V-\tilde{V}$，我们基于网格速度的光滑性进行了收敛性分析。此外，对流速度的降低改善了显式时间步进的稳定性，而采用非守恒的ALE公式则改变了强制条件。最后，通过将现有鲁棒误差准则适配于这种移动网格情境，我们推导出鲁棒的\textit{后验}误差准则，该准则描述了相对于平均流的潜在微小偏差，并包含了向$V=\tilde{V}$过渡的信息。