Convergence of Lowest Order Semi-Lagrangian Schemes

We consider generalized linear transient advection-diffusion problems for differential forms on a bounded domain in ℝd. We provide comprehensive a priori convergence estimates for their spatiotemporal discretization by means of a first-order in time semi-Lagrangian approach combined with a discontinuous Galerkin method. Under rather weak assumptions on the velocity underlying the advection we establish an asymptotic L2-estimate of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(\tau+h^{r}+h^{r+1}\tau^{-\frac{1}{2}}+\tau^{\frac{1}{2}})$\end{document}, where h is the spatial meshwidth, τ denotes the time step, and r is the polynomial degree of the forms used as trial functions. This estimate can be improved considerably in a variety of special settings.