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Sobolev meets Besov: Regularity for the Poisson equation with Dirichlet, Neumann and mixed boundary values

Cornelia Schneider

Friedrich-Alexander Universität Erlangen, Applied Mathematics III, Cauerstr. 11, 91058 Erlangen, Germany

E-mail Address: [email protected]

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and

Flóra O. Szemenyei

Friedrich-Alexander Universität Erlangen, Applied Mathematics III, Cauerstr. 11, 91058 Erlangen, Germany

E-mail Address: [email protected]

Corresponding author.

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https://doi.org/10.1142/S0219530522500026Cited by:2 (Source: Crossref)

Abstract

We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones $K \subset ℝ^{3}$ in the specific scale $B_{τ, τ}^{α}, \frac{1}{τ} = \frac{α}{d} + \frac{1}{p}$ of Besov spaces. The regularity of the solution in these spaces determines theorder of approximation that can be achieved by adaptive and nonlinear numerical schemes. We aim for a thorough discussion of homogeneous and inhomogeneous boundary data in all settings studied and show that the solutions are much smoother in this specific Besov scale compared to the fractional Sobolev scale $H^{s}$ in all cases,which justifies the use of adaptive schemes.

Keywords:

AMSC: 35B65, 46E35, 35J05, 35J25, 42C40, 65M12

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