Abstract
We study direct first-kind boundary integral equations arising from transmission problems for the Helmholtz equation with piecewise constant coefficients and Dirichlet boundary conditions imposed on a closed surface. We identify necessary and sufficient conditions for the occurrence of so-called spurious resonances, that is, the failure of the boundary integral equations to possess unique solutions.
Following [A. Buffa and R. Hiptmair, Numer Math, 100, 1–19 (2005)] we propose a modified version of the boundary integral equations that is immune to spurious resonances. Via a gap construction it will serve as the basis for a universally well-posed stabilized global multi-trace formulation that generalizes the method of [X. Claeys and R. Hiptmair, Commun Pure and Appl Math, 66, 1163–1201 (2013)] to situations with Dirichlet boundary conditions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- Ω i :
-
Material sub-domains \({\subset\mathbb{R}^{d}}\), Ω0 unbounded, see Fig. 1
- n :
-
Number of (bounded) sub-domains with penetrable medium
- \({\Sigma:=\partial\Omega_{\Sigma}}\) :
-
Boundary where homogeneous Dirichlet boundary conditions are imposed
- Γ:
-
Union of interfaces (skeleton), see (2.1)
- \({\gamma^{j}_{\rm D}}\), \({\gamma^{j}_{\rm N}}\) :
-
Dirichlet and Neumann trace operators on \({\partial\Omega_{j}}\), see (2.1)
- γj :
-
Cauchy trace operator defined in (2.5)
- \({\mathbb{H}(\partial\Omega_{j})}\) :
-
Cauchy trace space associated with \({\partial\Omega_{j}}\), see (3.1)
- \({\mathbb{H}(\Gamma)}\) :
-
Multi-trace space as defined in (3.1)
- \({{\langle \cdot, \cdot \rangle}_j}\) :
-
Duality pairing between Dirichlet and Neumann traces on \({\partial\Omega_{j}}\)
- \({{[\cdot, \cdot]}}\) :
-
Self-duality pairing on \({\mathbb{H}(\Gamma)}\)
- \({\mathbb{X}^{\pm\frac{1}{2}}(\Gamma)}\), \({\mathbb{X}(\Gamma)}\) :
-
Single trace Dirichlet/Neumann/Cauchy spaces, see (3.5), (3.6)
- \({{\sf T}_{\rm D}}\), \({{\sf T}_{\rm N}}\), \({{\sf T}}\) :
-
Restriction of single trace functions onto Σ, See Propositions (3.1), (3.2)
- \({{\sf SL}_{\kappa}^{j}}\) :
-
Single layer potential defined on \({\partial\Omega_{j}}\)
- \({{\sf DL}_{\kappa}^{j}}\) :
-
Double layer potential defined on \({\partial\Omega_{j}}\)
- \({{\sf G}_{\kappa}^{j}}\) :
-
Total potential defined on \({\partial\Omega_{j}}\)
- \({\mathcal{C}_{\kappa}(\partial\Omega_{j})}\) :
-
Space of Cauchy data on \({\partial\Omega_{j}}\)
- \({{\sf A}_{\kappa_{j}}^{j}}\) :
-
Boundary integral operator on \({\partial\Omega_{j}}\)
- \({{\sf B}_{i}{j}}\) :
-
Non-local “remote” coupling boundary integral operators
- \({\mathbb{X}_{0}(\Gamma)}\) :
-
Single trace space with vanishing Dirichlet data on Σ, see (4.1)
References
Brakhage H., Werner P.: Ueber das Dirichletsche Außenraumproblem für die Helmholtzsche Schwingungsgleichung. Arch. der Math. 16, 325–329 (1965)
Buffa A., Hiptmair R.: A coercive combined field integral equation for electromagnetic scattering. SIAM J. Numer. Anal. 42, 621–640 (2004)
Buffa A., Hiptmair R.: Regularized combined field integral equations. Numer. Math. 100, 1–19 (2005)
Buffa A., Hiptmair R., von Petersdorff T., Schwab C.: Boundary element methods for Maxwell transmission problems in Lipschitz domains. Numer. Math. 95, 459–485 (2003)
Buffa A., Sauter S.: On the acoustic single layer potential: stabilization and fourier analysis. SIAM J. Sci. Comput. 28, 1974–1999 (2006)
Burton A., Miller G.: The application of integral methods for the numerical solution of boundary value problems. Proc. R. Soc. Lond. Ser. A 232, 201–210 (1971)
Chang Y., Harrington R.: A surface formulation or characteristic modes of material bodies. IEEE Trans. Antennas Propag. 25, 789–795 (1977)
Christiansen S., Nédélec J.-C.: Des préconditionneurs pour la résolution numérique des équations intégrales de frontiére de l’acoustique. C.R. Acad. Sci. Paris Ser. I Math. 330, 617–622 (2000)
Claeys, X.: A single trace integral formulation of the second kind for acoustic scattering, Tech. Rep. no. 2011–2014, SAM, ETH Zürich (2011)
Claeys, X., Hiptmair, R.: Electromagnetic scattering at composite objects : a novel multi-trace boundary integral formulation. ESAIM: Math. Model Numer. Anal., 46, pp. 1421–1445
Claeys X., Hiptmair R.: Multi-trace boundary integral formulation for acoustic scattering by composite structures. Commun. Pure Appl. Math. 66, 1163–1201 (2013)
Claeys, X., Hiptmair, R., Jerez-Hanckes, C.: Multi-trace boundary integral equations. In: Graham, I., Langer, U., Melenk, J., Sini, M. (eds.) Direct and Inverse Problems in Wave Propagation and Applications, vol. 14 of Radon Series on Computational and Applied Mathematics. pp. 51–100, De Gruyter, Berlin/Boston (2013)
Claeys, X., Hiptmair, R., Jerez-Hanckes, J., Pintarelli, S.: Novel multi-trace boundary integral equations for transmission boundary value problems. Report 2014-XX, SAM, ETH Zürich, Switzerland (2014)
Colton, D., Kress, R.: Integral equation methods in scattering theory, Pure and Applied Mathematics (New York), John Wiley & Sons Inc. New York, A Wiley-Interscience Publication (1983)
Colton D., Kress R.: Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences. 2nd edn. Springer, Heidelberg (1998)
Costabel M., Stephan M.: A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106, 367–413 (1985)
Demkowicz L.: Asymptotic convergence in finite and boundary element methods: Part 1, Theoretical results. Comput. Math. Appl. 27, 69–84 (1994)
Harrington R.: Boundary integral formulations for homogeneous material bodies. J. Electromagn. Waves Appl. 3, 1–15 (1989)
Hiptmair R.: Operator preconditioning. Comput. Math. Appl. 52, 699–706 (2006)
Hiptmair R., Jerez-Hanckes C.: Multiple traces boundary integral formulation for Helmholtz transmission problems. Adv. Appl. Math. 37, 39–91 (2012)
Hiptmair, R., Jerez-Hanckes, C., Lee, J.-F., Peng Z.: Domain decomposition for boundary integral equations via local multi-trace formulations, Report 2013–08, SAM, ETH Zürich, 2013. In: Proceedings of XXI International Conference on Domain Decomposition Methods, Rennes, France, June 25–29 (2012)
Hiptmair R., Meury P.: Stabilized FEM–BEM coupling for Helmholtz transmission problems. SIAM J. Numer. Anal. 44, 2107–2130 (2006)
Hiptmair, R., Meury, P.: Stabilized FEM–BEM coupling for Maxwell transmission problems. In: Ammari, H. (ed.) Modelling and Computations in Electromagnetics, vol. 59 of Springer Lecture Notes in Computational Science and Engineering ch. 1. pp. 1–39, Springer, Berlin (2007)
Leis R.: Zur Dirichletschen Randwertaufgabe des Aussenraumes der Schwingungsgleichung. Math. Z. 90, 205–211 (1965)
Leis, R.: Initial boundary Value Problems in Mathematical Physics. In: Teubner, B.G., Stuttgart (eds.) Wiley, Chichester (1986)
McLean W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Nédélec J.-C.: Acoustic and electromagnetic Equations: Integral representations For Harmonic Problems, vol. 44 of Applied Mathematical Sciences. Springer, Berlin (2001)
Panich O.: On the question of the solvability of the exterior boundary-value problems for the wave equation and Maxwell’s equations. Usp. Mat. Nauk. 20, 221–226 (1965) In Russian
Peng Z., Lim K.-H., Lee J.-F.: Computations of electromagnetic wave scattering from penetrable composite targets using a surface integral equation method with multiple traces. IEEE Trans. Antennas Propag. 61, 256–270 (2013)
Peng Z., Wang X.-C., Lee J.-F.: Integral equation based domain decomposition method for solving electromagnetic wave scattering from non-penetrable objects. IEEE Trans. Antennas Propag. 59, 3328–3338 (2011)
Poggio A., Miller E.: Integral equation solution of three-dimensional scattering problems. In: Mittra, R. Computer Techniques for Electromagnetics, ch. 4, pp. 159–263. Pergamon, New York (1973)
Sauter S., Schwab C.: Boundary Element Methods, vol. 39 of Springer Series in Computational Mathematics. Springer, Heidelberg (2010)
Steinbach, O.: Integral equations for helmholtz boundary value and transmission problems. In: Graham, I., Langer, U., Melenk, J., Sini, M. (eds.) Direct and Inverse Problems in Wave Propagation and Applications, vol. 14 of Radon Series on Computational and Applied Mathematics, pp. 253–292, De Gruyter, Berlin/Boston (2013)
Steinbach O., Wendland W.: The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math. 9, 191–216 (1998)
Steinbach O., Windisch M.: Modified combined field integral equations for electromagnetic scattering. SIAM J. Numer. Anal. 47, 1149–1167 (2009)
Steinbach O., Windisch M.: Stable boundary element domain decomposition methods for the Helmholtz equation. Numer. Math. 118, 171–195 (2011)
von Petersdorff T.: Boundary integral equations for mixed Dirichlet, Neumann and transmission problems. Math. Method. Appl. Sci. 11, 185–213 (1989)
Wendland, W.: Boundary element methods for elliptic problems. In: Schatz, A., Thomée, V., Wendland, W. (eds.) Mathematical Theory of Finite and Boundary Element Methods, vol. 15 of DMV-Seminar, Birkhäuser, Basel, pp. 219–276 (1990)
Wu T.-K., Tsai L.-L.: Scattering from arbitrarily-shaped lossy dielectric bodies of revolution. Radio Sci. 12, 709–718 (1977)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work received financial support from Fondation ISAE, and from the French Ministry of Defense via DGA-MRIS.
Rights and permissions
About this article
Cite this article
Claeys, X., Hiptmair, R. Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects. Integr. Equ. Oper. Theory 81, 151–189 (2015). https://doi.org/10.1007/s00020-014-2197-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-014-2197-y