Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

Numerical Methods for Partial Differential Equations

Volumn 27, Issue 1, 2011, Pages 31-69

  Betcke, T ^a   Chandler Wilde, S N ^a   Graham, I G ^b   Langdon, S ^a   Lindner, M ^c  

^a UNIVERSITY OF READING   (United Kingdom)

^b University of Bath   (United Kingdom)

^c CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

boundary element method; Helmholtz equation; high oscillation

Indexed keywords

BOUNDARY ELEMENT METHOD; EIGENVALUES AND EIGENFUNCTIONS; HELMHOLTZ EQUATION; INTEGRAL EQUATIONS; SAILING VESSELS;

COMBINED FIELD INTEGRAL EQUATIONS; CONTINUOUS OPERATORS; DOUBLE LAYER POTENTIAL; HIGH OSCILLATION; LOWER AND UPPER BOUNDS; NUMERICAL EXPERIMENTS; POTENTIAL INTEGRALS; UPPER AND LOWER BOUNDS;

NUMBER THEORY;

EID: 78649846595     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20643     Document Type: Article

References (35)

1. D. L. Colton, and, R. Kress, Integral equation methods in scattering theory, John Wiley, New York, 1983.
2. W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.
3. S. N. Chandler-Wilde, and, S. Langdon, A Galerkin boundary element method for high frequency scattering by convex polygons, SIAM J Numer Anal 45 (2007), 610-640.
4. S. Amini, On the choice of the coupling parameter in boundary integral equation formulations of the exterior acoustic problem, Appl Anal 35 (1990), 75-92.
5. S. Amini, Boundary integral solution of the exterior acoustic problem, Comput Mech 13 (1993), 2-11.
6. L. Banjai, and, S. Sauter, A refined Galerkin error and stability analysis for highly indefinite variational problems, SIAM J Numer Anal 45 (2007), 37-53.
7. S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and, M. Lindner, Condition number estimates for combined potential boundary integral operators in acoustic scattering, J Int Eqn Appl 21 (2009), 229-279.
8. S. N. Chandler-Wilde, and, P. Monk, Wave-number-explicit bounds in time-harmonic scattering, SIAM J Math Anal 39 (2008), 1428-1450.
9. V. Dominguez, I. G. Graham, and, V. P. Smyshlyaev, A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering, Numer Math 106 (2007), 471-510.
10. 3, PhD Thesis, Universität Karlsruhe, Germany, 1997.
11. R. Kress, Minimizing the condition number of boundary integral-operators in acoustic and electromagnetic scattering, Q J Mech Appl Math 38 (1985), 323-341.
12. R. Kress, and, W. T. Spassov, On the condition number of boundary integral operators for the exterior Dirichlet problem for the Helmholtz equation, Numer Math 42 (1983), 77-95.
13. J. B. Keller, Semiclassical mechanics, SIAM Rev 27 (1985), 485-504.
14. J. Toth, Eigenfunction decay estimates in the quantum integrable case, Duke Math J 93 (1998), 231-255.
15. M. Löhndorf, and, J. M. Melenk, Wavenumber-explicit hp-BEM for high frequency scattering, ASC Report 2/2010, Institute for Analysis and Scientific Computing, Vienna University of Technology-TU Wien, 2010.
16. J. M. Melenk, Mapping properties of combined field Helmholtz boundary integral operators, ASC Report 1/2010, Institute for Analysis and Scientific Computing, Vienna University of Technology-TU Wien, 2010.
17. T. Betcke, and, E. A. Spence, Numerical estimation of coercivity constants for boundary integral operators in acoustic scattering, SIAM J Numer Anal, Submitted.
18. K. F. Warnick, and, W. C. Chew, Convergence of moment-method solutions of the electric field integral equation for a 2-D open cavity, Microwave Opt Tech Lett 23 (1999), 212-218.
19. K. F. Warnick, and, W. C. Chew, On the spectrum of the electric field integral equation and the convergence of the moment method, Int J Numer Meth Eng 51 (2001), 31-56.
20. K. F. Warnick, and, W. C. Chew, Error analysis of the moment method, IEEE Ant Prop Mag 46 (2004), 38-53.
21. O. P. Bruno, and, A. L. Kunyansky, Surface scattering in three dimensions: an accelerated high-order solver, Proc R Soc Lond A 457 (2001), 2921-2934.
22. O. P. Bruno, Private communication (2007).
23. M. Abramowitz, and, I. Stegun, Handbook of mathematical functions, Dover, New York, 1972.
24. V. F. Lazutkin, Semiclassical asymptotics of eigenfunctions, Encyclopedia of mathematical sciences, M. V. Fedoryuk, editor, Vol. 34, Partial differential equations V, Springer, New York, 1999, p. 133.
25. D. L. Colton, and, R. Kress, Inverse acoustic and electromagnetic scattering theory, 2nd Ed., Springer, New York, 1992.
26. R. Kress, Linear integral equations, 2nd Edition, Springer, New York, 1999.
27. G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J Funct Anal 59 (1984), 572-611.
28. S. Prössdorf, and, B. Silbermann, Numerical analysis for integral and related operator equations, Birkhäuser, Basel, 1991.
29. 1 domains, Acta Math 141 (1978), 165-186.
30. H. B. Wilson, and, R. W. Scharstein, Computing elliptic membrane high frequencies by Mathieu and Galerkin methods, J Eng Math 57 (2007), 41-55.
31. Digital Library of Mathematical Functions, Release date 2010-05-07, National Institute of Standards and Technology. Available at: http://dlmf.nist.gov/.
32. N. W. McLachlan, Theory and application of Mathieu functions, Oxford University Press, Oxford, 1947.
33. A. G. M. Neves, Eigenmodes and eigenfrequencies of vibrating elliptic membranes: a Klein oscillation theorem and numerical calculations, Comm Pure Appl Anal 9 (2010), 611-624.
34. S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations, Princeton University Press, Princeton NJ, 1982.
35. P. D. Hislop, and, I. M. Segal, Introduction to spectral theory: with applications to Schrödinger operators, Springer, New York, 1995.