On the efficient use of the Galerkin-method to solve Fredholm integral equations

Wolfgang Hackbusch; Stefan A. Sauter

Applications of Mathematics (1993)

  • Volume: 38, Issue: 4-5, page 301-322
  • ISSN: 0862-7940

Abstract

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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 + 1 n ) 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.

How to cite

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Hackbusch, Wolfgang, and Sauter, Stefan A.. "On the efficient use of the Galerkin-method to solve Fredholm integral equations." Applications of Mathematics 38.4-5 (1993): 301-322. <http://eudml.org/doc/15757>.

@article{Hackbusch1993,
abstract = {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.},
author = {Hackbusch, Wolfgang, Sauter, Stefan A.},
journal = {Applications of Mathematics},
keywords = {boundary element method; Galerkin method; numerical cubature; panel-clusterig-algorithm; Fredholm integral equations; numerical test; boundary integral equations; hypersingular kernels; splines; nearly singular integrals; error analysis; collocation method; Fredholm integral equations; boundary element method; numerical test; boundary integral equations; hypersingular kernels; splines; nearly singular integrals; Galerkin method; panel-clustering method; error analysis; collocation method},
language = {eng},
number = {4-5},
pages = {301-322},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the efficient use of the Galerkin-method to solve Fredholm integral equations},
url = {http://eudml.org/doc/15757},
volume = {38},
year = {1993},
}

TY - JOUR
AU - Hackbusch, Wolfgang
AU - Sauter, Stefan A.
TI - On the efficient use of the Galerkin-method to solve Fredholm integral equations
JO - Applications of Mathematics
PY - 1993
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 38
IS - 4-5
SP - 301
EP - 322
AB - 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.
LA - eng
KW - boundary element method; Galerkin method; numerical cubature; panel-clusterig-algorithm; Fredholm integral equations; numerical test; boundary integral equations; hypersingular kernels; splines; nearly singular integrals; error analysis; collocation method; Fredholm integral equations; boundary element method; numerical test; boundary integral equations; hypersingular kernels; splines; nearly singular integrals; Galerkin method; panel-clustering method; error analysis; collocation method
UR - http://eudml.org/doc/15757
ER -

References

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  12. S. Sauter, Über die effiziente Verwandung des Galerkinverfahrens zur Lösung Fredholmscher Intergleichungen, Dissertation, Universität Kiel, 1992. (1992) 
  13. C. Schwab W. Wendland, Kernel Properties and Representations of Boundary Integral Operators, Preprint 91-92, Universität Stuttgart, to appear in Math. Nachr.. MR1233945
  14. C. Schwab W. Wendland, On numerical cubatures of singular surface integrals in boundary element methods, Num. Math. (1992), 343-369. (1992) MR1169009
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