Time-dependent problems often require different resolution in different spatial regions, time intervals, and polynomial directions. Tensor-product space-time finite element methods provide a framework in which these anisotropies can be represented directly by local mesh refinement and anisotropic hp enrichment in space and time. This additional flexibility, however, leads to large coupled systems and therefore requires suitable solvers.
We present anisotropic hp-adaptive space-time finite element methods and multilevel preconditioners for the resulting linear systems. The adaptive strategy is based on goal-oriented error control in the Dual Weighted Residual framework. Directional error indicators identify whether refinement is most relevant in space, in time, or in the local polynomial degrees. This yields discretizations that adapt to goal-relevant structures without enforcing isotropic refinement.
The preconditioners exploit the Kronecker structure of tensor-product space-time discretizations. They decouple the temporal degrees of freedom for continuous and discontinuous time discretizations, leading to shifted spatial subproblems coupled through low-dimensional temporal factors. This makes the linear systems arising from anisotropic hp-adaptive space-time methods computationally feasible.
Numerical examples for convection-dominated transport problems, the heat equation, and the Stokes equations illustrate the interaction between anisotropic hp adaptivity and multilevel preconditioning.