FoCM 2014 conference

Workshop A5 - Multiresolution and Adaptivity in Numerical PDEs

December 11, 17:30 ~ 17:50 - Room B22

## Regularity of boundary integral equations in Besov-type spaces based on wavelet expansions

### Markus Weimar

### Philipps-University Marburg, Germany - weimar@mathematik.uni-marburg.de

We study regularity properties of solutions to operator equations on patchwise smooth manifolds $\partial\Omega$ such as, e.g., boundaries of polyhedral domains $\Omega \subset \mathbb{R}^3$. Using suitable biorthogonal wavelet bases $\Psi$, we introduce a new class of Besov-type spaces $B_{\Psi,q}^\alpha(L_p(\partial \Omega))$ of functions $u\colon\partial\Omega\rightarrow\mathbb{C}$. 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 $\partial\Omega$ into $B_{\Psi,\tau}^\alpha(L_\tau(\partial \Omega))$, $1/\tau=\alpha/2 + 1/2$, 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 $\Omega$. The talk is based on two recent papers which arose from the DFG-Project ''BIOTOP: Adaptive Wavelet and Frame Techniques for Acoustic BEM'' (DA 360/19-1):

Dahlke, S. and Weimar, M.: Besov regularity for operator equations on patchwise smooth manifolds. Preprint 2013-03, Fachbereich Mathematik und Informatik, Philipps-Universität Marburg. To appear in J. Found. Comput. Math.

Weimar, M.: Almost diagonal matrices and Besov-type spaces based on wavelet expansions. Preprint 2014-06, Fachbereich Mathematik und Informatik, Philipps-Universität Marburg. Submitted.

Joint work with Stephan Dahlke (Philipps-University Marburg, Germany).