Adaptive refinement for the simplex space-time finite element method

A large share of physical problems encountered in engineering applications is time-dependent. To solve those problems numerically, the space-time (ST) finite element method (FEM), obtained by employing a finite element discretization in space and time, can be used. By benefiting from the single discretization of the entire space-time domain, adaptive refinement in space and time can be applied to cut computational costs or increase the accuracy. Therefore, we show an adaptive mesh refinement (AMR) procedure in space and time on conforming simplex space-time (SST) discretizations driven by combinations of recovery- and jump-based error estimators. A recovery-based error estimator yields error estimates and indicators in space or time based on the chosen gradient contributions. The jump-based error estimator, driven by the weakly enforced continuity between space-time slabs, provides error indicators for temporal refinement. Based on the local error indicators, locally refined or coarsened space-time slabs are obtained. We then apply the error estimators and refinement procedure to a time-dependent advection-diffusion problem.