Geodesic
On the efficient use of the Galerkin-method to solve Fredholm integral equations
Applications of Mathematics, Tome 38 (1993) no. 4-5, pp. 301-322

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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 $O(n \log^k^+^1n)$ 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.
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.
DOI : 10.21136/AM.1993.104558
Classification : 35J25, 45B05, 45E05, 45E10, 65D30, 65D32, 65N38, 65R20
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 
Hackbusch, Wolfgang; Sauter, Stefan A. On the efficient use of the Galerkin-method to solve Fredholm integral equations. Applications of Mathematics, Tome 38 (1993) no. 4-5, pp. 301-322. doi: 10.21136/AM.1993.104558
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