Research

Research interests

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.

Key aspects:

  • 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.

Function spaces

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.

Key aspects:

  • 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

Projects

  • 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)
    Abstract
    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.

  • 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)
    Abstract
    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.