Stephan Dahlke

Petru A. Cioica, Stephan Dahlke, Stefan Kinzel et al.

We use the scale of Besov spaces B^α_{τ,τ}(O), α>0, 1/τ=α/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.

Stephan Dahlke, Lars Diening, Christoph Hartmann et al.

In this paper, we study the regularity of solutions to the $p$-Poisson equation for all $1<p<\infty$. In particular, we are interested in smoothness estimates in the adaptivity scale $ B^σ_τ(L_τ(Ω))$, $1/τ= σ/d+1/p$, of Besov spaces. The regularity in this scale determines the order of approximation that can be achieved by adaptive and other nonlinear approximation methods. It turns out that, especially for solutions to $p$-Poisson equations with homogeneous Dirichlet boundary conditions on bounded polygonal domains, the Besov regularity is significantly higher than the Sobolev regularity which justifies the use of adaptive algorithms. This type of results is obtained by combining local Hölder with global Sobolev estimates. In particular, we prove that intersections of locally weighted Hölder spaces and Sobolev spaces can be continuously embedded into the specific scale of Besov spaces we are interested in. The proof of this embedding result is based on wavelet characterizations of Besov spaces.

Stephan Dahlke, Markus Weimar

We study regularity properties of solutions to operator equations on patchwise smooth manifolds $\partialΩ$ such as, e.g., boundaries of polyhedral domains $Ω\subset \mathbb{R}^3$. Using suitable biorthogonal wavelet bases $Ψ$, we introduce a new class of Besov-type spaces $B_{Ψ,q}^α(L_p(\partial Ω))$ of functions $u\colon\partialΩ\rightarrow\mathbb{C}$. Special attention is paid on the rate of convergence for best $n$-term wavelet approximation to functions in these scales since this determines the performance of adaptive numerical schemes. We show embeddings of (weighted) Sobolev spaces on $\partialΩ$ into $B_{Ψ,τ}^α(L_τ(\partial Ω))$, $1/τ=α/2 + 1/2$, which lead us to regularity assertions for the equations under consideration. Finally, we apply our results to a boundary integral equation of the second kind which arises from the double layer ansatz for Dirichlet problems for Laplace's equation in $Ω$.

Stephan Dahlke, Helmut Harbrecht, Manuela Utzinger et al.

In this paper, we are concerned with the numerical treatment of boundary integral equations by means of the adaptive wavelet boundary element method (BEM). In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in $\mathbb{R}^3$. The corresponding operator equations are treated by means of adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions that can be made on the basis of a systematic investigation of the Besov regularity of the exact solution. Keywords: Besov spaces, weighted Sobolev spaces, adaptive wavelet BEM, non-linear approximation, integral equations, double layer potential operator, regularity, manifolds.

Stephan Dahlke, Erich Novak, Winfried Sickel

We study the optimal approximation of the solution of an operator equation by certain n-term approximations with respect to specific classes of frames. We study worst case errors and the optimal order of convergence and define suitable nonlinear frame widths. The main advantage of frames compared to Riesz basis, which were studied in our earlier papers, is the fact that we can now handle arbitrary bounded Lipschitz domains--also for the upper bounds. Key words: elliptic operator equation, worst case error, frames, nonlinear approximation, best n-term approximation, manifold width, Besov spaces on Lipschitz domains