Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings III: Frames

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 domainsComment: J. Complexity, to appear. Final version, minor mistakes correcte

Besov regularity of solutions to the p-Poisson equation

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 $B^\sigma_{\tau}(L_{\tau}(\Omega))$, $1/\tau = \sigma/d+1/p$, 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 H\"older with global Sobolev estimates. In particular, we prove that intersections of locally weighted H\"older 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.Comment: 45 pages, 2 figure