Abstract
Based on a direct method proposed by Christiansen and Hougaard we consider spline approximation solution of the clamped plate problem. Our discretization methods cover the Galerkin and collocation solutions. The system of boundary equations is not strongly elliptic. However we are able to derive optimal order error estimates for unknown boundary densities in some Sobolev spaces. The corresponding asymptotic convergence for approximation of the biharmonic function itself has been observed also in numerical calculations.
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
Arnold D.N. and Wendland W.L. (1983), On the Asymptotic Convergence of Collocation Methods, Math. Compo 41, pp. 349–381.
Arnold D.N. and Wendland W.L. (1985), The Convergence of Spline Collocation for Strongly Elliptic Equations on Curves, Numer. Math. 47, pp. 317–341.
Christiansen S. and Hougaard P. (1978), An Investigation of a Pair of Integral Equations for the Biharmonic Problem, J. Inst. Math. Appl. 22, pp. 15–27.
Costabel M., Lusikka L. and Saranen J., Comparison of Three Boundary Element Approaches for the Solution of the Clamped Plate Problem, Boundary Elements IX. 2. Southampton, 1987, pp. 19–34.
Costabel M., Stephan E. and Wendland W.L. (1983), On Boundary Integral Equations of the First Kind for the Bi-Laplacian in a Polygonal Plane Domain, Ann. Scuola Norm. Sup. Pisa, C1. Sci. (4), 10, pp. 197–241.
Costabel M. and Wendland W. L. (1986), Strong Ellipticity of Boundary Integral Operators, J. Reine Angew. Math. 372, pp. 39–63.
Dieudonne J. (1978). Elements d’Analyse Vol. 8, Gauthier-Villars. Paris.
Elschner J. and Schmidt G. (1983), On Spline Interpolation in Periodic Sobolev Spaces, Preprint 01/83, Dept. Math. Akademie der Wissenschaften der DDR, Berlin.
Fuglede B. (1981), On a Direct Method of Integral Equations for Solving the Biharmonic Dirichlet Problem, ZAMM 61, pp. 449–459.
Hsiao G.C. and MacCamy R. (1973), Solution of Boundary Value Problems by. Integral Equations of the First Kind, SIAM Rev. 15, pp. 687–705.
Hsiao G.C. and Wendland W.L. (1981), The Aubin-Nitsche Lemma for Integral Equations, J. Integral Equations 3, pp. 299–315.
Lusikka 1., Ruotsalainen K. and Saranen J. (1986), Numerical Implementationof the Boundary Element Method with Point-Source Approximation of the Potential, Engineering Analysis, Vol. 3, No. 3, pp. 144–153.
Ruotsalainen K. and Saranen J., Some Boundary Element Methods Using Dirac’s Distributions as Trial Functions, SIAM J. Numer. Anal. 24, 1987, pp. 816–827.
Ruotsalainen K. and Saranen J., A Dual Method to the Collocation Method, to appear in Math. Meth. Appl. Sci.
Ruotsalainen K. and Saranen J., On the Convergence of the Galerkin Method for Nonsmooth Solutions of Integral Equations, to appear in Numerische Mathematik.
Saranen J., The Convergence of Even Degree Spline Collocation Solution for Potential Problems in Smooth Domains of the Plane, to appear in Numerische Mathematik.
Saranen J. and Wendland W.L. (1985), On the Asymptotic Convergence of Collocation Methods with Spline Functions of Even Degree, Math. Compo 45, pp. 91–108.
Wendland W.L. (1981), Asymptotic Convergence of Boundary Element Methods, in Lectures on the Numerical Solution of Partial Differential Equations (Ed. Babuska 1., Liu T.P. and Osborn J.), pp. 435–528, Univ. of Maryland, College Park Md, 198!.
Wendland W.L. (1983), Boundary Element Method and their Asymptotic Convergence, in Theoretical Acoustics and Numerical Techniques (Ed. Filippi P.), pp. 135–216, CISM Courses 277, Springer-Verlag, 1983. Wien and New York.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Prof. I. Gohberg on the occasion of his sixtieth birthday
Rights and permissions
Copyright information
© 1989 Birkhäuser Verlag Basel
About this chapter
Cite this chapter
Costabel, M., Saranen, J. (1989). Boundary Element Analysis of a Direct Method for the Biharmonic Dirichlet Problem. In: Dym, H., Goldberg, S., Kaashoek, M.A., Lancaster, P. (eds) The Gohberg Anniversary Collection. Operator Theory: Advances and Applications, vol 40/41. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9144-8_24
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9144-8_24
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9924-6
Online ISBN: 978-3-0348-9144-8
eBook Packages: Springer Book Archive