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

Abstract

We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones [Formula: see text] in the specific scale [Formula: see text] 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 [Formula: see text] in all cases,which justifies the use of adaptive schemes.