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Besov regularity of edge singularities for the Poisson equation in polyhedral domains
Author(s) -
Dahlke Stephan
Publication year - 2002
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.304
Subject(s) - mathematics , besov space , gravitational singularity , bounded function , singularity , domain (mathematical analysis) , pure mathematics , mathematical analysis , mathematical proof , scale (ratio) , interpolation space , geometry , quantum mechanics , gene , biochemistry , chemistry , physics , functional analysis
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.