zbMATH — the first resource for mathematics

Fast cluster techniques for BEM. (English) Zbl 1035.65142
Summary: We present a new approach for solving boundary integral equations with panel clustering. In contrast to all former versions of panel clustering, the computational and storage complexity of the algorithm scales linearly with respect to the number of degrees of freedom without any additional logarithmic factors. The idea is to develop alternative formulations of all classical boundary integral operators for the Laplace problem where the kernel function has a reduced singular behaviour. It turns out that the application of the panel-clustering method with variable approximation order preserves the asymptotic convergence rate of the discretisation and has significantly reduced complexity.
Reviewer: Reviewer (Berlin)

65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
Full Text: DOI
[1] Börm S, Sauter S. Alternative representations of classical boundary integral operators allowing cheap quadrature formulae and low order panel-clustering approximations. In preparation.
[2] Erichsen, S.; Sauter, S., Efficient automatic quadrature in 3-d Galerkin BEM, Comput meth appl mech engng, 157, 215-224, (1998) · Zbl 0943.65139
[3] Graham, I.; Hackbusch, W.; Sauter, S., Hybrid Galerkin boundary elements: theory and implementation, Numer math, 86, 98-6, 139-172, (2000) · Zbl 0966.65091
[4] Hackbusch, W., Elliptic differential equations, (1992), Springer Berlin
[5] Hackbusch, W., Integral equations, (1995), ISNM Birkhäuser
[6] Hackbusch, W.; Nowak, Z., On the fast matrix multiplication in the boundary element method by panel-clustering, Numer math, 54, 463-491, (1989) · Zbl 0641.65038
[7] Hackbusch, W.; Sauter, S., On the efficient use of the Galerkin method to solve Fredholm integral equations, Appl math, 38, 4-5, 301-322, (1993) · Zbl 0791.65101
[8] McLean, W., Strongly elliptic systems and boundary integral equations, (2000), Cambridge University Press Cambridge · Zbl 0948.35001
[9] Nédélec, J., Integral equations with non-integrable kernels, Integ eq oper theor, 5, 562-572, (1982) · Zbl 0479.65060
[10] Nédélec, J., Acoustic and electromagnetic equations, (2001), Springer Berlin
[11] Rokhlin, V., Rapid solutions of integral equations of classical potential theory, J comput phys, 60, 2, 187-207, (1985) · Zbl 0629.65122
[12] Sauter, S., Variable order panel clustering, Computing, 64, 223-261, (2000) · Zbl 0959.65135
[13] Sauter, S.; Lage, C., Transformation of hypersingular integrals and black-box cubature, Math comput, 70, 97-17, 223-250, (2001) · Zbl 0958.65123
[14] Sauter, S.; Schwab, C., Quadrature for hp-Galerkin BEM in \(R\^{}\{3\}\), Numer math, 78, 2, 211-258, (1997) · Zbl 0901.65069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.