**Third party funded individual grant**

**Start date :**
01.04.2020

**End date :**
30.09.2021

In this project we study parabolic partial differential equations (=PDEs) on bounded Lipschitz domains. We aim at justifying the use of adaptive numerical methods when treating such equations. In an adaptive strategy the choice of the underlying degrees of freedom is not a priori fixed but depends on the shape of the unknown solutions. Additional degrees of freedom are only spent in regions where the numerical approximation is still far away from the exact solution. The best one can expect from an adaptive algorithm is an optimal performance in the sense that it realizes the convergence rate of best N-term approximation (i.e., best approximation of the solution by linear combinations with at most N basis functions). However, this convergence order depends on the regularity of the solution in specific scales of Besov spaces. It is therefore our aim to investigate the Besov regularity of the solutions of parabolic PDEs in order to see whether adaptivity pays off in this context. In the first funding period of the project we were able to show that adaptivity is indeed justified for quite general classes of linear and nonlinear parabolic PDEs. Even better regularity results could be achieved for polyhedral cones (instead of general Lipschitz domains). In the second funding period of the project we wish to improve and develop these results further. Our achievements for cones have to be generalized to polyhedral domains. Moreover, the nonlinear results on the Besov regularity so far are only established on convex domains. Since from a numerical point of view non-convex domains are of particular interest, we want to prove similar results here. Furthermore, we plan to study the regularity in fractional Sobolev spaces for stochastic parabolic PDEs, which determines the convergence order of non-adaptive methods. Also an investigation of the Besov regularity of PDEs on more general manifolds (e.g. soap films) is intended. As another aspect we study the approximation classes of parabolic PDEs. Our goal here is a convergence analysis of the horizontal mothod of lines (Rothe's method), when we use a Galerkin-method for our discretization in time and adaptive discretizations in space. Moreover, instead of a time-marching algorithm (as described above) we could use a full space-time adaptive algorithm based on tensor wavelets. Numerical studies indicate that this is more efficient. In particular, the approximation order that can be achieved this way turns out to be independent of the spatial dimansion and depends on the regulairity of the exact solution in a specific scale of tensor products of Besov spaces. Therefore, we will systematically investigate the regularity of the solutions in these scales of dominating mixed smoothness spaces.