Convergence of lowest-order semi-Lagrangian schemes (Q1955528)

The authors consider a non-stationary advection-diffusion problem for time-dependent differential forms. By means of the Hille-Yosida theorem, the existence and the uniqueness of the transient advection-diffusion problem are obtained. The semi-Lagrangian Galerkin time-stepping scheme for the considered advection-diffusion problem is presented. Under some additional assumptions, an \(L^2\)-estimate of order \(O(\tau+h^r+h^{r+1}\tau^{-1/2}+\tau^{1/2})\) is established, with \(h\) the spatial meshsize, \(\tau\) the time step and \(r\) the polynomial degree of the trial functions.