| Title:
|
On the efficient use of the Galerkin-method to solve Fredholm integral equations (English) |
| Author:
|
Hackbusch, Wolfgang |
| Author:
|
Sauter, Stefan A. |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
38 |
| Issue:
|
4 |
| Year:
|
1993 |
| Pages:
|
301-322 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the present paper we describe, how to use the Galerkin-method efficiently in solving boundary integral equations. In the first part we show how the elements of the system matrix can be computed in a reasonable time by using suitable coordinate transformations. These techniques can be applied to a wide class of integral equations (including hypersingular kernels) on piecewise smooth surfaces in 3-D, approximated by spline functions of arbitrary degree.
In the second part we show, how to use the panel-clustering technique for the Galerkin-method. This technique was developed by Hackbusch and Nowak in [6,7] for the collocation method. In that paper it was shown, that a matrix-vector-multiplication can be computed with a number of $O(n \log^k^+^1n)$ operations by storing $O(n \log^k n)$ sizes. For the panel-clustering-techniques applied to Galerkin-discretizations we get similar asymptotic estimates for the expense, while the reduction of the consumption for practical problems (1 000-15 000 unknowns) turns out to be stronger than for the collocation method. (English) |
| Keyword:
|
boundary element method |
| Keyword:
|
Galerkin method |
| Keyword:
|
numerical cubature |
| Keyword:
|
panel-clusterig-algorithm |
| Keyword:
|
Fredholm integral equations |
| Keyword:
|
numerical test |
| Keyword:
|
boundary integral equations |
| Keyword:
|
hypersingular kernels |
| Keyword:
|
splines |
| Keyword:
|
nearly singular integrals |
| Keyword:
|
error analysis |
| Keyword:
|
collocation method |
| MSC:
|
35J25 |
| MSC:
|
45B05 |
| MSC:
|
45E05 |
| MSC:
|
45E10 |
| MSC:
|
65D30 |
| MSC:
|
65D32 |
| MSC:
|
65N38 |
| MSC:
|
65R20 |
| idZBL:
|
Zbl 0791.65101 |
| idMR:
|
MR1228511 |
| DOI:
|
10.21136/AM.1993.104558 |
| . |
| Date available:
|
2008-05-20T18:46:04Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104558 |
| . |
| Reference:
|
[1] M. Costabel W. L. Wendland: Strong ellipticity of boundary integral operators.J. Reine Angew. Math., 1986. MR 0863517 |
| Reference:
|
[2] M. Costabel E. P. Stephan W. L. Wendland: On boundary integral equations of the first kind for the bi-Laplacian in a polygonal domain.Ann. Sc. Norm. Sup. Pisa, Classe di Scienze, Serie IV X (1983), no. 2. |
| Reference:
|
[3] A. Friedman: Partial Differential Equations.Holt, Rinehart and Winston, Inc. New York, 1969. Zbl 0224.35002, MR 0445088 |
| Reference:
|
[4] W. Hackbusch: Multi-grid methods and Applications.Springer-Verlag, Berlin, 1985. Zbl 0595.65106 |
| Reference:
|
[5] W. Hackbusch: Integralgleichungen.Teubner, Stuttgart, 1989. Zbl 0681.65099, MR 1010893 |
| Reference:
|
[6] W. Hackbusch Z. P. Nowak: O: n the complexity of the panel method.in the proceedings of the conference "Modern Problems in Numerical Analysis", Moscow, Sept. 1986. (In Russian.) |
| Reference:
|
[7] W. Hackbusch Z. P. Nowak: On the fast matrix multiplication in the boundary element method by panel-clustering.Num. Math. 54 (1989), 436-491. MR 0972420 |
| Reference:
|
[8] F. John: Plane waves and spherical means.Springer-Verlag, New York, 1955. Zbl 0067.32101 |
| Reference:
|
[9] Z. P. Nowak: Efficient panel methods for the potential flow problems in the three space dimensions.Report Nr. 8815, Universitat Kiel, 1988. |
| Reference:
|
[10] N. Ortner: Construction of Fundamental Solutions.Topics in Boundary Element Research (C. A. Brebbia, ed.), to appear. |
| Reference:
|
[11] S. Sauter: Der Aufwand der Panel-Clustering-Methode für Integralgleichungen.Report Nr. 9115, Universität Kiel, 1991. |
| Reference:
|
[12] S. Sauter: Über die effiziente Verwandung des Galerkinverfahrens zur Lösung Fredholmscher Intergleichungen.Dissertation, Universität Kiel, 1992. |
| Reference:
|
[13] C. Schwab W. Wendland: Kernel Properties and Representations of Boundary Integral Operators.Preprint 91-92, Universität Stuttgart, to appear in Math. Nachr.. MR 1233945 |
| Reference:
|
[14] C. Schwab W. Wendland: On numerical cubatures of singular surface integrals in boundary element methods.Num. Math. (1992), 343-369. MR 1169009 |
| Reference:
|
[15] W. Wendland: Boundary element methods and their asymptotic convergence.Theoretical Acoustics and Numerical Treatments (P. Filippi, ed.), Pentech Press, London, Plymouth, 1981, pp. 289-313. |
| Reference:
|
[16] W. Wendland: Asymptotic Accuracy and Convergence for Point Collocation Methods.Topics in Boundary Element Research, Vol. 2 (C. A. Brebbia, ed.), Springer-Verlag, Berlin, 1985, pp. 230-257. Zbl 0597.65085, MR 0823729 |
| Reference:
|
[17] W. L. Wendland: Strongly elliptic boundary integral equations.The State of the Art in Numerical Analysis (A. Iserles and M. Powell, eds.), Clarendon Press, Oxford, 1987, pp. 511-561. Zbl 0615.65119, MR 0921677 |
| . |
