Regularity theory of partial differential equations
My research is motivated by the following fundamental question: In which cases do adaptive algorithms outperform non-adaptive (uniform) numerical schemes? In general the convergence order for non-adaptive algorithms is determined by the regularity of the exact solution of the underlying equation in the scale of (classical) Sobolev spaces. In contrast to this a theoretical analysis shows, that the achievable approximation order for adaptive schemes depends on the regularity of the target function in a special scale of Besov spaces. Therefore, in order to justify the use of adaptive schemes, an analysis of the regularity of the solution in the scale of Besov spaces and a comparison with its Sobolev regularity is needed.
- Regularity theory for nonlinear elliptic equations
- Regularity theory for (nonlinear) parabolic equations
- Regularity theory for ellipic equations on manifolds
- Regularity theory for hyperbolic equations
Adaptive schemes for parabolic problems
The goal is to extend already existing adaptive strategies for elliptic problems to the case of parabolic equations. In order to do this suitable time-marching algorithms need to be developed. In every time-step one makes use of a fully adaptive elliptic solver in space.
Concerning the theory of function spaces my research is centered around general Besov and Sobolev-type spaces. Typical problems arising in this context are different approaches and characterizations of function spaces (in particular on various settings such as manifolds, domains, fractals), their connections and diversities as well as embeddings between various scales and traces.
- Besov-, Sobolev- and Triebel-Lizorkin spaces
- Kondratiev spaces (weighted Sobolev spaces)
- Smoothness Morrey spaces
- Function spaces with variable exponents
- Characterizations (e.g. via atoms, wavelets, higher differences)
- Embeddings, traces
Besov regularity of parabolic partial differential equations on Lipschitz domains (continuation)
(Third Party Funds Single)Term: 01-04-2020 - 30-09-2021
Funding source: Deutsche Forschungsgemeinschaft (DFG)
Besov Regularität von parabolischen partiellen Differentialgleichungen auf Lipschitz Gebieten
(Third Party Funds Single)Term: 01-04-2017 - 31-03-2019
Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
This project is concerned with partial differential equations (PDEs) of parabolic type on Lipschitz domains. We want to study the regularity of the solutions of such equations and are particularly interested in smoothness estimates in specific scales of Besov spaces, which determine the approximation order of adaptive and other nonlinear numerical approximation schemes. We wish to show that the Besov regularity is high enough to justify the use of adaptive schemes compared to non-adaptive (uniform) schemes. We shall mainly be concerned with approximation schemes based on wavelets.Our starting point is to improve existing results for the heat equation and extend these results to linear parabolic PDEs with variable coefficients and also to nonlinear parabolic PDEs. We also want to show that when restricting ourselves to polygonal and polyhedral domains (compared to general Lipschitz domains) we get even higher Besov regularity.
Since the approximation rates obtained this way still depend on the spatial dimension, we secondly wish to study regularity of parabolic PDEs in generalized dominating mixed smoothness spaces. By using tensor-wavelets we want to beat the so-called 'curse of dimension' and obtain convergence rates independent of the underlying dimension of the domain.