## Efficient automatic quadrature in 3-D Galerkin BEM.(English)Zbl 0943.65139

Summary: We present cubature methods approximating the surface integrals arising by Galerkin discretization of boundary integral equations on surfaces in $$\mathbb{R}^3$$. This numerical integrator does not depend on the explicit form of the kernel function, the trial and test space, or the surface parametrization. Thus, it is possible to generate the system matrix for a broad class of integral equations just by replacing the subroutine for evaluating the kernel function. We present formulae to determine the minimal order of the cubature methods for a required accuracy. Emphasis is laid on numerical experiments confirming the theoretical results.

### MSC:

 65N38 Boundary element methods for boundary value problems involving PDEs 65D32 Numerical quadrature and cubature formulas 35J25 Boundary value problems for second-order elliptic equations
Full Text:

### References:

 [1] Hackbusch, W., Integral equations. theory and numerical treatment, (1995), Birkhaeuser-Verlag Basel, Boston, Berlin · Zbl 0823.65139 [2] Hackbusch, W.; Sauter, S.A., On the efficient use of the Galerkin method to solve Fredholm integral equations, Appl. math., 38, 4-5, 301-322, (1993) · Zbl 0791.65101 [3] Lage, C., Softwareentwicklung zur randelementmethode: analyse und entwurf effizienter techniken, () [4] Nedelec, J.C., Integral equations with non-integrable kernels, Integral eqns. oper. theory, 5, 562-572, (1982) · Zbl 0479.65060 [5] Sauter, S.A.; Krapp, A., On the effect of numerical integration in the Galerkin boundary element method, Numer. math., 74, 3, 337-359, (1996) · Zbl 0878.65104 [6] Sauter, S.A.; Lage, C., On the efficient computation of singular and nearly singular surface integrals arising in 3D-Galerkin BEM, Zamm, 76, 2, 273-275, (1996) · Zbl 0886.65141 [7] S.A. Sauter and C. Schwab, Quadrature for hp-Galerkin BEM in 3-d, Numer. Math., to appear. · Zbl 0901.65069 [8] Sauter, S.A., Über die effiziente verwendung des galerkinverfahrens zur Lösung fredholmscher integralgleichungen, () · Zbl 0850.65366 [9] Sauter, S.A.; Schwab, C., Realization of hp-Galerkin BEM in 3-d, (), 194-206 · Zbl 0881.65115 [10] Schwab, C.; Wendland, W., Kernel properties and representations of boundary integral operators, Math. nachr., 156, 187-216, (1992) · Zbl 0805.35168 [11] Stroud, A.H., Approximate calculation of multiple integrals, (1971), Prentice-Hall Englewood Cliffs, NJ · Zbl 0379.65013 [12] Petersdorff, T.v.; Schwab, C., Full discrete multiscale Galerkin BEM, (), to appear [13] Wendland, W.L., Boundary element methods and their asymptotic convergence, (), 289-313 · Zbl 0618.65109
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.