Hostname: page-component-6bb9c88b65-lm65w Total loading time: 0.006 Render date: 2025-07-25T10:14:24.592Z Has data issue: false hasContentIssue false
  • English
  • Français

Piecewise-constant collocation for first-kind boundary integral equations

Published online by Cambridge University Press:  17 February 2009

I. G. Graham  and
Y. Yan
I. G. Graham
Affiliation:
School of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom.
Y. Yan
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky, U.S.A.
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.

We examine the piecewise-constant collocation method, with collocation points the mid-points of subintervals, for first-kind integral equations with logarithmic kernels on polygonal boundaries. Previously this method had been shown to converge subject to certain restrictions on the angles at the corners of the polygon. Here, by considering a slightly modified collocation method, we are able to remove any restrictions on these angles, and to generalise slightly the meshes which may be used. Moreover, the modification leads to new results on the convergence of preconditioned two-(or multi-) grid methods for solving the resultant linear systems.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 33 , Issue 1 , July 1991 , pp. 39 - 64
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Atkinson, K. E. and de Hoog, F. R., “The numerical solution of Laplace's equation on a wedge”, IMA J. Numer. Anal. 4 (1984) 19–41.CrossRefGoogle Scholar
[2]Atkinson, K. E. and Graham, I. G., “An iterative variant of the Nystrom method for boundary integral equations on nonsmooth boundaries”, in The Mathematics of Finite Elements and Applications (ed. Whiteman, J. R.), (Academic Press, London, 1988).Google Scholar
[3]Arnold, D. N. and Wendland, W. L., “The convergence of spline collocation for strongly elliptic equations on curves”, Numer. Math. 47 (1985), 317–341.CrossRefGoogle Scholar
[4]Chan, R. H. and Strang, G., “Toeplitz equations by conjugate gradients with circulant preconditioner”, SI AM J. Sci. Stat. Comp. 10 (1989) 104–119.Google Scholar
[5]Chandler, G. A., “Numerical analysis of the boundary integral method”, in Mathematical programming and numerical analysis workshop (eds. Gustafson, S. A. and Womersley, R. S.), (Proceedings of the Centre for Mathematical Analysis, Australian National University, v. 6, 1983).Google Scholar
[6]Chandler, G. A. and Graham, I. G., “Product integration–collocation methods for non-compact integral operator equations”, Math. Comp. 50 (1988) 125–138.CrossRefGoogle Scholar
[7]Christiansen, S., “On two methods for elimination of non-unique solutions of an integral equation with logarithmic kernel”, Applicable Analysis 13 (1982) 1–18.CrossRefGoogle Scholar
[8]Costabel, M. and Stephan, E. P., “On the convergence of collocation methods for boundary integral equations on polygons”, Math. Comp. 49 (1987) 467–478.CrossRefGoogle Scholar
[9]Costabel, M. and Stephan, E. P., “Boundary integral equations for mixed boundary value roblems in polygonal domains and Galerkin approximation”, in Mathematical models nd methods in mechanics, (Banach Centre Publications 15, PWN, Warsaw, 1985) 175–251.Google Scholar
[10]De Hoog, F. R., “Product integration techniques for the numerical solution of integral equations”, Ph. D. Thesis, Australian National University, 1973.Google Scholar
[11]Hackbusch, W., Multigrid methods and applications (Springer-Verlag, Berlin, 1985).CrossRefGoogle Scholar
[12]Hebeker, F.-K., “On multigrid methods of the first kind for symmetric boundary integral equations of nonnegative order”, Preprint No. 1120, Fachbereich Mathematik, Technische Hochschule Darmstadt, 1988.Google Scholar
[13]Hsiao, G. C. and Wendland, W. L., “A finite element method for some integral equations of the first kind”, J. Math. Anal. Applies. 58 (1977) 449–481.CrossRefGoogle Scholar
[14]Jaswon, M. A. and Symm, G. T., Integral equation methods in potential theory and elastostatics, (Academic Press, London, 1977).Google Scholar
[15]McLean, W., “Boundary integral methods for the Laplace equation”, Ph. D. Thesis, Australian National University, 1985.Google Scholar
[16]Saranen, J., “The convergence of even degree spline collocation solution for potential problems in smooth domains of the plane”, Numer. Math. 52 (1988) 499–512.CrossRefGoogle Scholar
[17]Saranen, J and Wendland, W. L., “On the asymptotic convergence of collocation methods with spline functions of even degree”, Math. Comp. 45 (1985), 91–108.CrossRefGoogle Scholar
[18]Schippers, H., “Multignd methods for boundary integral equations”, Numer. Math. 46 (1985) 351–363.CrossRefGoogle Scholar
[19]Symm, G. T., “Integral equation methods in potential theory II”, Proc. Roy. Soc. (A) 275 (1963) 33–46.Google Scholar
[20]Yan, Y. and Sloan, I. H., “Mesh grading for integral equations of the first kind with logarithmic kernel’, SIAMJ. Numer. Anal. 26 (1989) 574–587.CrossRefGoogle Scholar
[21]Yan, Y. and Sloan, I. H., “On integral equations of the first kind with logarithmic kernels”, J. Integral Equations and Applications 1 (1988) 549–579.CrossRefGoogle Scholar
[22]Yan, Y., “The collocation method for first kind boundary integral equations on polygonal domains”, Math. Comp. 54 (1990) 139–154.CrossRefGoogle Scholar