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Foundations of Computational Mathematics
Erschienen in: Foundations of Computational Mathematics 6/2015

01.12.2015

Besov regularity for operator equations on patchwise smooth manifolds

verfasst von: Stephan Dahlke, Markus Weimar

Erschienen in: Foundations of Computational Mathematics | Ausgabe 6/2015

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Abstract

We study regularity properties of solutions to operator equations on patchwise smooth manifolds \(\partial \Omega \), 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 \(u:\partial \Omega \rightarrow \mathbb {C}\). 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 \(\Omega \).
Vorheriger Artikel Metrics for Generalized Persistence Modules
Nächster Artikel Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach

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Metadaten
Titel
Besov regularity for operator equations on patchwise smooth manifolds
verfasst von
Stephan Dahlke
Markus Weimar
Publikationsdatum
01.12.2015
Verlag
Springer US
Erschienen in
Foundations of Computational Mathematics / Ausgabe 6/2015
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-015-9273-9

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