Space–time discontinuous Galerkin method for advection–diffusion problems on time-dependent domains

This article presents a space–time discontinuous Galerkin (DG) finite element discretization of the advection–diffusion equation on time-dependent domains. In the space–time DG discretization no distinction is made between the space and time variables and discontinuous basis functions are used both in space and time. This approach results in an efficient numerical technique for physical applications which require moving and deforming elements, is suitable for hp-adaptation and results in a fully conservative discretization. A complete derivation of the space–time DG method for the advection–diffusion equation is given, together with the relation of the space–time discretization with the arbitrary Lagrangian Eulerian (ALE) approach. Detailed proofs of stability and error estimates are also provided. The space–time DG method is demonstrated with numerical experiments that agree well with the error analysis.