A velocity-based moving mesh Discontinuous Galerkin method for the advection-diffusion equation
<|reference_start|>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 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 $V=\tilde{V}$.<|reference_end|>
@article{rozier2024a,
    title={A velocity-based moving mesh Discontinuous Galerkin method for the
  advection-diffusion equation},
    author={Ezra Rozier and J"orn Behrens},
    journal={arXiv preprint arXiv:2405.09408},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.09408},
    primaryClass={math.NA cs.NA}
}