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

Erichsen, Stefan; Sauter, Stefan A.

Comput. Methods Appl. Mech. Eng. 157, No. 3-4, 215-224 (1998).

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

Keywords:

automatic quadrature; Fredholm integral equations; boundary element method; cubature methods; surface integrals; Galerkin discretization; boundary integral equations; numerical experiments

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