Error-controlled space-time finite elements, algorithms and implementations for nonstationary problems

Abstract

Recently, many advances have been done in the field of space-time finite element discretizations for nonstationary partial differential equations. The temporal dimension leads to additional costs in solv- ing the equations, so adaptive finite element methods can lead to sufficiently accurate solutions at a better efficiency compared to uniform refinement. Additionally, the solution to the equation itself is often not of direct interest, but instead some derived quantity has to be calculated accurately. For this, the dual-weighted residual (DWR) method offers a practical way to calculate estimators for the temporal and spatial error parts with respect to the unknown exact value of that quantity. This thesis introduces the extension of the partition-of-unity localization (PU) for non-stationary prob- lems, which provides error indicators for use in marking strategies. As the choice and application of this PU differs slightly for different choices of the space-time discretization approach we give a short overview over the three choices and the resulting (discrete) function spaces. Additionally, we look at some more technical details of what a finite element library needs to offer to be able to apply one of these approaches. For the tensor-product approach we developed the library ideal.II which extends the library deal.II to simplify the implementation of non-stationary problems. We discuss some of the design decisions and technical details of this library. Both the library and the PU-DWR method are validated by several numerical studies with linear and nonlinear PDEs.