# 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

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topHackbusch, 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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