Fully discrete semi-Lagrangian methods for advection of differential forms

We study the discretization of linear transient transport problems for differential forms on bounded domains. The focus is on unconditionally stable semi-Lagrangian methods that employ finite element approximation on fixed meshes combined with tracking of the flow map. We derive these methods as finite element Galerkin approach to discrete material derivatives and discuss further approximations leading to fully discrete schemes. We establish comprehensive a priori error estimates, in particular a new asymptotic estimate of order O(h^r+1 τ ^-1/2) for the L²-error of semi-Lagrangian schemes with exact L²-projection. Here, h is the spatial meshwidth, τ denotes the timestep, and r is the (full) polynomial degree of the piecewise polynomial discrete differential forms used as trial functions. Yet, numerical experiments hint that the estimates may still be sub-optimal for spatial discretization with lowest order discrete differential forms.