On the efficient use of the Galerkin-method to solve Fredholm integral equations. (English) Zbl 0791.65101

Hackbusch, Wolfgang; Sauter, Stefan A.

Appl. Math., Praha 38, No. 4-5, 301-322 (1993).

Let \(\Gamma\) be a piecewise smooth two-dimensional manifold in \(\mathbb{R}^ 3\). The authors consider boundary integral equations of the form \[ \lambda (x)u(x)+Ku(x)=f(x),\quad x \in \Gamma, \tag{1} \] where \[ Ku(x):=\int_ \Gamma k(x,y,y-x) u(y)d \Gamma(y),\;k(x,y,y-x)={1 \over | y-x |^{s+t}} \sum_{| \alpha | \geq t} c_ \alpha (x,y)(y-x)^ \alpha, \] with \(s+t\) odd, \(t \in N_ 0\) and piecewise smooth functions \(c_ \alpha\). This covers a wide class of integral equations (including hypersingular kernels) on piecewise smooth surfaces, approximated by splines of arbitrary degree. Using suitable coordinate transformations, the computation of the singular and nearly singular integrals arising in the discretization of (1) via the Galerkin method is discussed.
Furthermore, the panel-clustering method introduced by the first author and Z. P. Nowak [Numer. Math. 54, No. 4, 463-491 (1989; Zbl 0652.65031)] is developed for Galerkin discretizations and the corresponding error analysis is presented, yielding similar asymptotic estimates for the expense as shown in that paper for the collocation method. Numerical test calculations show that the reduction of the panel- clustering method is stronger for the Galerkin procedure than for the collocation method.

Reviewer: J.Elschner (Berlin)

MSC:

65R20	Numerical methods for integral equations
65N38	Boundary element methods for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations
45E10	Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)

Keywords:

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

Citations:

Zbl 0641.65038; Zbl 0652.65031

References:

[1]	M. Costabel W. L. Wendland: Strong ellipticity of boundary integral operators. J. Reine Angew. Math., 1986. · Zbl 0628.35027
[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. · Zbl 0563.45007
[3]	A. Friedman: Partial Differential Equations. Holt, Rinehart and Winston, Inc. New York, 1969. · Zbl 0224.35002
[4]	W. Hackbusch: Multi-grid methods and Applications. Springer-Verlag, Berlin, 1985. · Zbl 0595.65106
[5]	W. Hackbusch: Integralgleichungen. Teubner, Stuttgart, 1989. · Zbl 0681.65099
[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. · Zbl 0646.76031
[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. · Zbl 0641.65038
[8]	F. John: Plane waves and spherical means. Springer-Verlag, New York, 1955. · Zbl 0067.32101
[9]	Z. P. Nowak: Efficient panel methods for the potential flow problems in the three space dimensions. Report Nr. 8815, Universitat Kiel, 1988.
[10]	N. Ortner: Construction of Fundamental Solutions. Topics in Boundary Element Research (C. A. Brebbia, to appear. · Zbl 1043.35007
[11]	S. Sauter: Der Aufwand der Panel-Clustering-Methode für Integralgleichungen. Report Nr. 9115, Universität Kiel, 1991.
[12]	S. Sauter: Über die effiziente Verwandung des Galerkinverfahrens zur Lösung Fredholmscher Intergleichungen. Dissertation, Universität Kiel, 1992.
[13]	C. Schwab W. Wendland: Kernel Properties and Representations of Boundary Integral Operators. Preprint 91-92, Universität Stuttgart, to appear in Math. Nachr.. · Zbl 0805.35168
[14]	C. Schwab W. Wendland: On numerical cubatures of singular surface integrals in boundary element methods. Num. Math. (1992), 343-369. · Zbl 0761.65012
[15]	W. Wendland: Boundary element methods and their asymptotic convergence. Theoretical Acoustics and Numerical Treatments (P. Filippi, Pentech Press, London, Plymouth, 1981, pp. 289-313.
[16]	W. Wendland: Asymptotic Accuracy and Convergence for Point Collocation Methods. Topics in Boundary Element Research, Vol. 2 (C. A. Brebbia, Springer-Verlag, Berlin, 1985, pp. 230-257. · Zbl 0597.65085
[17]	W. L. Wendland: Strongly elliptic boundary integral equations. The State of the Art in Numerical Analysis (A. Iserles and M. Powell, Clarendon Press, Oxford, 1987, pp. 511-561. · Zbl 0615.65119