Previous |  Up |  Next


boundary element method; Galerkin discretization; Helmholtz equation; hypersingular boundary integral equation
We deal with the Galerkin discretization of the boundary integral equations corresponding to problems with the Helmholtz equation in 3D. Our main result is the semi-analytic integration for the bilinear form induced by the hypersingular operator. Such computations have already been proposed for the bilinear forms induced by the single-layer and the double-layer potential operators in the monograph The Fast Solution of Boundary Integral Equations by O. Steinbach and S. Rjasanow and we base our computations on these results.
[1] Grigorieff, R. D., Sloan, I. H.: Galerkin approximation with quadrature for the screen problem in $\mathbb{R}^3$. J. Integral Equations Appl. 9 (1997), 293-319. DOI 10.1216/jiea/1181076026 | MR 1614302
[2] Mauersberger, D., Sloan, I. H.: A simplified approach to the semi-discrete Galerkin method for the single-layer equation for a plate. M. Bonnet, et al. Mathematical Aspects of Boundary Element Methods Minisymposium during the IABEM 98 conference France, 1998, Chapman Hall, Boca Raton. Notes Math. 414 178-190 (2000). MR 1719844 | Zbl 0937.65142
[3] McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press Cambridge (2000). MR 1742312 | Zbl 0948.35001
[4] Nédélec, J.-C.: Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems. Applied Mathematical Sciences 144 Springer, New York (2001). DOI 10.1007/978-1-4757-4393-7_3 | MR 1822275 | Zbl 0981.35002
[5] Of, G., Steinbach, O., Wendland, W. L.: The fast multipole method for the symmetric boundary integral formulation. IMA J. Numer. Anal. 26 (2006), 272-296. DOI 10.1093/imanum/dri033 | MR 2218634 | Zbl 1101.65114
[6] Rjasanow, S., Steinbach, O.: The Fast Solution of Boundary Integral Equations. Mathematical and Analytical Techniques with Applications to Engineering Springer, New York (2007). MR 2310663 | Zbl 1119.65119
[7] Sauter, S., Schwab, C.: Boundary Element Methods. Springer Series in Computational Mathematics 39 Springer, Berlin (2011). DOI 10.1007/978-3-540-68093-2 | MR 2743235 | Zbl 1215.65183
[8] Zapletal, J.: The Boundary Element Method for the Helmholtz Equation in 3D. MSc. thesis, Department of Applied Mathematics, VŠB-TU, Ostrava (2011).
Partner of
EuDML logo