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Graded Mesh Refinement and Error Estimates for Finite Element Solutions of Elliptic Boundary Value Problems in Non‐smooth Domains
Author(s) -
Apel Thomas,
Sändig AnnaMargarete,
Whiteman John R.
Publication year - 1996
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19960110)19:1<63::aid-mma764>3.0.co;2-s
Subject(s) - mathematics , finite element method , lipschitz continuity , gravitational singularity , context (archaeology) , convergence (economics) , sobolev space , polygon mesh , boundary value problem , conical surface , boundary (topology) , rate of convergence , mathematical analysis , geometry , computer science , paleontology , channel (broadcasting) , computer network , physics , biology , economics , thermodynamics , economic growth
This paper is concerned with the effective numerical treatment of elliptic boundary value problems when the solutions contain singularities. The paper deals first with the theory of problems of this type in the context of weighted Sobolev spaces and covers problems in domains with conical vertices and non‐intersecting edges, as well as polyhedral domains with Lipschitz boundaries. Finite element schemes on graded meshes for second‐order problems in polygonal/polyhedral domains are then proposed for problems with the above singularities. These schemes exhibit optimal convergence rates with decreasing mesh size. Finally, we describe numerical experiments which demonstrate the efficiency of our technique in terms of ‘actual’ errors for specific (finite) mesh sizes in addition to the asymptotic rates of convergence.