Semi-Lagrangian Galerkin schemes are popular methods for solving transient advection-diffusion problems due to their unconditional stability. They employ finite element approximation on a fixed mesh combined with tracking of the flow map. That procedure introduces finite element functions on distorted meshes that need to be projected onto finite element functions on the fixed mesh. The convergence analysis of fully discrete semi-Lagrangian Galerkin schemes is delicate.
In this talk we will focus on convergence of fully discrete semi-Lagrangian schemes for the magnetic advection-diffusion problem and lowest order N\'ed\'elec elements. Classical \emph{uniform} error bounds for semi-Lagrangian methods for the scalar advection-diffusion problems are of the type $O(\tau + h^{r+1}\tau^{-1})$, where $r$ is the polynomial degree of the approximation space, $\tau$ is the time step length, and $h$ is the characteristic mesh width. In the scalar case we always have $r\geq 1$, but approximation spaces for vectorial advection-diffusion problems may fail to contain all piecewise linear vector fields, which means $r=0$. This is exactly the case for lowest-order N\'ed\'elec elements and the magnetic advection-diffusion. For this simplest choice of trial spaces numerical analysis tools known so far completely fail to predict convergence. We present a new technique that uses a special auxiliary Galerkin discretization of a related stationary advection-diffusion problem to obtain improved error estimates.
Joint work with Ralf Hiptmair