Abstract
This paper is concerned with the anisotropic singular behaviour of the solution of elliptic boundary value problems in domains with edges and its consequences for anisotropic FEM. We first deal with the description of the analytic properties of the solution in newly defined anisotropic weighted Sobolev spaces. The finite element method with anisotropic, graded meshes and piecewise linear shape functions is then investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates in anisotropically weighted Sobolev spaces are derived.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Th. Apel and M. Dobrowolski, Anisotropic interpolation with applications to the finite element method. Computing, 47:277–293, 1992.
Th. Apel and B. Heinrich, Mesh refinement and windowing near edges for some elliptic problem. SIAM J. Numer. Anal., 31:695–708, 1994.
Th. Apel and S. Nicaise, Elliptic problems in domains with edges: Anisotropic regularity and anisotropic finite element meshes. Preprint SPC 94–16, TU Chemnitz-Zwickau, 1994.
Th. Apel, R. Mücke, and J. R. Whiteman, An adaptive finite element technique with a-priori mesh grading. Technical Report 9, BICOM Institute of Computational Mathematics, 1993.
Th. Apel, A.-M. Sändig, and J. R. Whiteman, Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Technical Report 12, BICOM Institute of Computational Mathematics, 1993. To appear in Math. Meth. Appl. Sci.
P. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Company, Amsterdam, 1978.
P. Grisvard, Singular behaviour of elliptic problems in non Hilbertian Sobolev spaces. J. Math. Pures Appl., 74:3–33, 1995.
V. A. Kondrat’ev, Singularities of the solution of the Dirichlet problem for a second order elliptic equation in the neighbourhood of an edge. Differencial’nye uravnenija, 13(11):2026–2032, 1977. (Russian).
M. S. Lubuma and S. Nicaise, Dirichlet problems in polyhedral domains II: approximation by FEM and BEM. J. Comp. Appl. Math., 61:13–27, 1995.
V. G. Maz’ya and B. A. Plamenevskiĭ, Lp-estimates of solutions of elliptic boundary value problems in domains with edges. Trudy Moskov. Mat. Obshch., 37:49–93, 1978. (Russian).
V. G. Maz’ya and J. Roßmann, Über die Asymptotik der Lösung elliptischer Randwertaufgaben in der Umgebung von Kanten. Math. Nachr., 138:27–53, 1988.
L. A. Oganesyan and L. A. Rukhovets, Variational-difference methods for the solution of elliptic equations. Izd. Akad. Nauk Armyanskoi SSR, Jerevan, 1979. (Russian).
G. Raugel, Résolution numérique de problèmes elliptiques dans des domaines avec coins. PhD thesis, Université de Rennes (France), 1978.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Birkhäuser Boston
About this paper
Cite this paper
Apel, T., Nicaise, S. (1996). Elliptic Problems in Domains with Edges: Anisotropic Regularity and Anisotropic Finite Element Meshes. In: Cea, J., Chenais, D., Geymonat, G., Lions, J.L. (eds) Partial Differential Equations and Functional Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 22. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2436-5_2
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2436-5_2
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-7536-7
Online ISBN: 978-1-4612-2436-5
eBook Packages: Springer Book Archive