Besov regularity of edge singularities for the Poisson equation in polyhedral domains
Stephan Dahlke · Numerical Linear Algebra with Applications · 2002
Abstract This paper is concerned with the regularity of the solutions to the Poisson equation in polyhedral domains Ω contained in R 3 . Especially, we consider the specific scale B τ α ( L τ (Ω)),1/τ = α/3 + ½, of Besov spaces. The regularity of the variational solution in these Besov spaces determines the order of approximation that can be achieved by adaptive and non‐linear numerical schemes. It is well‐known that in polyhedral domains different types of singularities according to edges and vertices occur. In this paper, we shall primarily be concerned with the Besov regularity of edge singularities. By the usual reduction procedure, these singularity functions are studied in an unfinite cylinder R × ϒ, where ϒ denotes a bounded polygonal domain contained in R 2 . We show that the singularity functions are much smoother in the specific Besov scale than in the usual L 2 –Sobolov scale which justifies the use of adaptive schemes. The proofs are based on specific representations of the solutions and on characterizations of Besov spaces by wavelet expansions. Copyright © 2002 John Wiley & Sons, Ltd.