An exact bounded perfectly matched layer for time-harmonic scattering problems

SIAM Journal on Scientific Computing

Volumn 30, Issue 1, 2007, Pages 312-338

  Bermudez A ^a   Hervella Nieto, L ^b   Prieto, A ^a   Rodriguez R ^c  

^a UNIVERSITY OF SANTIAGO DE COMPOSTELA   (Spain)

^b UNIVERSITY OF A CORUÑA   (Spain)

^c UNIVERSITY OF CONCEPCIÓN   (Chile)

Author keywords

Helmholtz equation; Perfectly matched layer; Time harmonic scattering

Indexed keywords

HARMONIC ANALYSIS; NUMERICAL METHODS; SOBOLEV SPACES;

HELMHOLTZ'S EQUATIONS; HIGH-ACCURACY; PERFECTLY MATCHED LAYER; SCATTERING PROBLEMS; STANDARD FINITE ELEMENT; TIME-HARMONIC; TIME-HARMONIC SCATTERING; UNBOUNDED DOMAIN; WEAK FORMULATION; WEIGHTED SOBOLEV SPACES;

HELMHOLTZ EQUATION;

EID: 44749088184     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/060670912     Document Type: Article

References (43)

1. S. ABARBANEL, D. GOTTLIEB, AND J. S. HESTHAVEN, Well-posed perfectly matched layers for advective acoustics, J. Comput. Phys., 154 (1999), pp. 266-283.
2. M. ABRAMOWITZ AND I. A. STEGUN, Handbook of Numerical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1970.
3. S. ASVADUROV, V. DRUSKIN, M. N. GUDDATI, AND L. KNIZHNERMAN, On optimal finitedifference approximation of PML, SIAM J. Numer. Anal., 41 (2003), pp. 287-305.
4. U. BASU AND A. K. CHOPRA, Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: Theory and finite-element implementation. Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 1337-1375.
5. A. BAYLISS AND E. TURKEL, Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math., 33 (1980), pp. 707-725.
6. E. BÉCACHE, A.-S. BONNET-BEN DHIA, AND G. LEGENDRE, Perfectly matched layers for the convected Helmholtz equation, SIAM J. Numer. Anal., 42 (2004), pp. 409-433.
7. E. BÉCACHE AND P. JOLY, On the analysis of Bérenger's perfectly matched layers for Maxwell's equations, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 87-119.
8. J. P. BERENGER, A perfectly matched layer for the absorption of electromagnetics waves, J. Comput. Phys., 114 (1994), pp. 185-200.
9. J. P. BERENGER, Perfectly matched layer for the FDTD solution of wave-structure interaction problems, IEEE Trans. Antennas Propag., 44 (1996), pp. 110-117.
10. J. P. BERENGER, Three-dimensional perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1996), pp. 363-379.
11. A. BERMÚDEZ, L. HERVELLA-NIETO, A. PRIETO, AND R. RODRÍGUEZ, An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems, J. Comput. Phys., 223 (2007), pp. 469-488.
12. H. BREZIS, Analyse fonctionnelle. Theorie et Applications, Mason, Paris, 1983.
13. Z. CHEN AND X. LIU, An adaptive perfectly matched layer technique for time-harmonic scattering problems, SIAM J. Numer. Anal., 43 (2005), pp. 645-671.
14. W. C. CHEW AND W. H. WEEDON, A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates, IEEE Microwave Opt. Technol. Lett., 7 (1994), pp. 599-604.
15. F. COLLINO AND P. MONK, The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput., 19 (1998), pp. 2061-2090.
16. F. COLLINO AND P. MONK, Optimizing the perfectly matched layer, Comput. Methods Appl. Mech. Engrg., 164 (1998), pp. 157-171.
17. F. COLLINO AND C. TSOGKA, Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media, Geophys., 66 (2001), pp. 294-1307.
18. D. COLTON AND R. KRESS, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, Heidelberg, 1992.
19. L. DEMKOWICZ AND F. IHLENBURG, Analysis of a coupled finite-infinite element method for exterior Helmholtz problems, Numer. Math., 88 (2001), pp. 43-73.
20. B. ENGQUIST AND A. MAJDA, Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31 (1977), pp. 629-651.
21. K. FENG, Finite element method and natural boundary reduction, in Proceedings of the International Congress of Mathematicians, Warsaw, Poland, 1983, pp. 1439-1453
22. D. GIVOLI, Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992.
23. I. HARARI, M. SLAVUTIN, AND E. TURKEL, Analytical and numerical studies of a finite element PML for the Helmholtz equation, J. Comput. Acoust., 8 (2000), pp. 121-137.
24. T. HOHAGE, F. SCHMIDT, AND L. ZSCHIEDRICH, Solving time-harmonic scattering problems based on the pole condition II: Convergence of the PML method, SIAM J. Math. Anal., 35 (2003), pp. 547-560.
25. F. Q. HU, A stable, perfectly matched layer for linearized Euler equations in unsplit physical variables, J. Comput. Phys., 173 (2001), pp. 455-480.
26. F. IHLENBURG, Finite Element of Analysis of Acoustical Scattering, Springer-Verlag, New York, 1998.
27. A. KUFNER AND A.-M. SÄNDIG, Some Applications of Weighted Sobolev Spaces, Texte zur Mathematik 100, Teubner, Leipzig, 1987.
28. M. LASSAS AND E. SOMERSALO, On the existence and convergence of the solution of PML equations, Computing, 60 (1998), pp. 228-241.
29. M. LASSAS AND E. SOMERSALO, Analysis of the PML equations in general convex geometry, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), pp. 1183-1207.
30. M. LASSAS, J. LIUKKONEN, AND E. SOMERSALO, Complex Riemannian metric and absorbing boundary conditions, J. Math. Pures Appl., 80 (2001), pp. 739-768.
31. J. L. LIONS AND E. MAGENES, Problémes aux Limites non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968.
32. J. L. LIONS, J. MÉTRAL, AND O. VACUS, Well-posed absorbing layer for hyperbolic problems, Numer. Math., 92 (2002), pp. 535-562.
33. M. MASMOUDI, Numerical solution for exterior problems, Numer. Math., 51 (1987), pp. 87-101.
34. W. MCLEAN, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, London, 2000.
35. I. M. NAVON, B. NETA, AND M. Y. HUSSAINI, A perfectly matched layer approach to the linearized shallow water equations models, Monthly Weather Rev., 132 (2004), pp. 1369-1378.
36. J. C. NEDELEC, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Springer-Verlag, New York, 2000.
37. Q. QI AND T. L. GEERS, Evaluation of the-perfectly matched layer for computational acoustics, J. Comput. Phys., 139 (1998), pp. 166-183.
38. C. M. RAPPAPORT, Interpreting and improving the PML absorbing boundary condition using anisotropic lossy mapping of space, IEEE Trans. Magnetics, 32 (1996), pp. 968-974.
39. J. J. SHIRRON AND I. BABUSKA, A comparison of approximate boundary conditions and infinite element methods for exterior Helmholtz problems, Comput. Methods Appl. Mech. Engrg., 164 (1998), pp. 121-139.
40. E. TURKEL AND A. YEFET, Absorbing PML boundary layers for wave-like equations, Appl. Numer. Math., 27 (1998), pp. 533-557.
41. G. N. WATSON, A Treatise on the Theory of Bessel Functions, Cambridge University Press, London, 1966.
42. C. H. WILCOX, Scattering Theory for the d'Alambert Equation in Exterior Domains, Springer-Verlag, Berlin, 1975.
43. Y. Q. ZENG, J. Q. HE, AND Q. H. LIU, The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media, Geophys., 66 (2001), pp. 1258-1266.