Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-29T11:12:13.437Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on the boundary element method for strongly nonlinear problems

Published online by Cambridge University Press:  17 February 2009

Keijo Ruotsalainen
Keijo Ruotsalainen
Affiliation:
University of Oulu, Faculty of Technology, Section of Mathematics, SF-90570 Oulu, Finland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Recently in several papers the boundary element method has been applied to non-linear problems. In this paper we extend the analysis to strongly nonlinear boundary value problems. We shall prove the convergence and the stability of the Galerkin method in Lp spaces. Optimal order error estimates in Lp space then follow. We use the theory of A-proper mappings and monotone operators to prove convergence of the method. We note that the analysis includes the u4 -nonlinearity, which is encountered in heat radiation problems.

Type
Research Article
Information
The ANZIAM Journal , Volume 34 , Issue 4 , April 1993 , pp. 419 - 438
Copyright
Copyright © Australian Mathematical Society 1993

References

[1] Adams, R. A., Sobolev spaces (Academic Press, New York, 1975).Google Scholar
[2] Arnold, D. and Wendland, W. L., “On the asymptotic convergence of collocation methods”, Math. Comp. 41 (1983) 349–381.CrossRefGoogle Scholar
[3] Babuška, I. and Aziz, A. K., “Survey lectures on the mathematical foundation of the finite element methods”, The mathematical foundation of the finite element method with applications to partial differential equations (Academic Press, New York, 1972) 3–359.Google Scholar
[4] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces (Noordhoff International Publishing, Leyden, 1978).Google Scholar
[5] Brézis, H., “Problemés unilatereaux”, J. Math. Pures et Appliquées 51 (1972) 1–168.Google Scholar
[6] Costabel, M., “Starke elliptizität von Randintegraloperatoren erster Art”, Habilitationsschrift Darmstadt 1984.Google Scholar
[7] Costabel, M. and Stephan, E., “Boundary integral eqautions for mixed boundary value problem in polygonal domains and Galerkin approximation”, Mathematical Models and Methods in Mechanics (Polish Scientific Publ. 1985).Google Scholar
[8] Boor, C. de, “A bound on the L∞-norm of L2 -approximation by splines in terms of global mesh ratio”, Math. Comp. 30 (1976) 765–771.Google Scholar
[9] Deimling, K., Nonlinear functional analysis (Springer-Verlag, Heidelberg, 1985).CrossRefGoogle Scholar
[10] Eggermont, P. and Saranen, J., “Lp estimates of boundary integral equations for some nonlinear boundary value problems” (to appear).Google Scholar
[11] Elschner, J. and Schmidt, G., “On spline interpolation in periodic Sobolev spaces”, P-Math-01/83 (Akad. Wiss. DDR, Inst. Math., Berlin 1983).Google Scholar
[12] Futik, S., Necas, J., Souček, J. and Souček, V., Spectral analysis of nonlinear operators (Springer Verlag, Berlin-Heidelberg-New York, 1973).Google Scholar
[13] Hackbusch, W. and Nowak, , “On the fast matrix multiplication in the boundary element method by panel clustering”, Numer. Math. 54 (1989) 463–491.CrossRefGoogle Scholar
[14] Hamina, M., Ruotsalainen, K. and Saranen, J., “Numerical approximation for the solution of a nonlinear boundary integral equation with the collocation method”, J. Int. Eqs. and Appl. 4 (1992) 1–21.Google Scholar
[15] Jörgens, K., Linear integral operators (Teubner, Stuttgart, 1970).CrossRefGoogle Scholar
[16] Oden, J. T., Qualitative methods in nonlinear mechanics (Prentice-Hall, New Jersey, 1986).Google Scholar
[17] Pascali, D. and Sburlan, S., Nonlinear mappings of monotone type (Siijthoff and Noordhoff International Publ., Bucarest, 1978).CrossRefGoogle Scholar
[18] Petryshyn, W. V., “On the approximation-solvability of equations involving A-proper and pseudo. A-proper mappings”, Bull. Amer. Math. Soc. 81 (1975) 223–312.CrossRefGoogle Scholar
[19] Ruotsalainen, K., “On the the boundary element method for a mixed non-linear boundary value problem”, Applicaple Analysis (to appear).Google Scholar
[20] Ruotsalainen, K. and Saranen, J., “On the collocation method for a nonlinear boundary integral equation”, J. Comp. Appl. Math. 28 (1989) 339–348.CrossRefGoogle Scholar
[21] Ruotsalainen, K. and Wendland, W. L., “On the boundary element method for some nonlinear boundary value problems”, Numer. Math. 53 (1988) 299–314.CrossRefGoogle Scholar
[22] Saranen, J., “Projection methods for a class of Hammerstein equations”, SIAM J. Numer. Anal, (to appear).Google Scholar
[23] Schechter, M., Principles of functional analysis (Academic Press, New York, 1971).Google Scholar
[24] Schumaker, L., Spline functions: Basic theory (John Wiley, New York, 1981).Google Scholar
[25] Vainberg, M. M., Variational methods for the study of nonlinear operators (Holden-Day, San Francisco, 1964).Google Scholar
[26] Verchota, G., “Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains”, J. Fund. Anal. 59 (1984) 572–611.CrossRefGoogle Scholar