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Optimized Schwarz Domain Decomposition Methods for Scalar and Vector Helmholtz Equations

  • X. Antoine
  • C. GeuzaineEmail author
Chapter
Part of the Geosystems Mathematics book series (GSMA)

Abstract

In this chapter we review Schwarz domain decomposition methods for scalar and vector Helmholtz equations, with a focus on the choice of the associated transmission conditions between the subdomains. The methods are analyzed in both acoustic and electromagnetic settings, and generic weak formulations directly amenable to finite element discretization are presented. An open source solver along with ready-to-use examples is freely available online for further testing.

Keywords

Domain Decomposition Transmission Condition Perfectly Match Layer Domain Decomposition Method Transmission Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work was supported in part by the Wallonia-Brussels Federation (ARC grant for Concerted Research Actions ARC WAVES 15/19-03), the Belgian Science Policy (PAI grant P7/02), the Walloon Region (WIST3 grants ONELAB and ALIZEES), the French ANR (grant MicroWave NT09 460489 “Programme Blanc”) and the “EADS Foundation” (High-BRID project, grant 089-1009-1006). Computational resources have been provided by CÉCI, funded by F.R.S.-FNRS (Fonds de la Recherche Scientifique) under grant n2. 5020. 11, and the Tier-1 supercomputer of the Fédération Wallonie-Bruxelles, funded by the Walloon Region under grant n1117545.

References

  1. 1.
    A. Alonso-Rodriguez and L. Gerardo-Giorda. New nonoverlapping domain decomposition methods for the harmonic Maxwell system. SIAM J. Sci. Comput., 28(1):102–122, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent. A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM Journal on Matrix Analysis and Applications, 23(1):15–41, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    X. Antoine, M. Darbas, and Y.Y. Lu. An improved surface radiation condition for high-frequency acoustic scattering problems. Comput. Methods Appl. Mech. Engrg., 195(33–36):4060–4074, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    X. Antoine, C. Geuzaine, and K. Ramdani. Wave Propagation in Periodic Media - Analysis, Numerical Techniques and Practical Applications, volume 1, chapter Computational Methods for Multiple Scattering at High Frequency with Applications to Periodic Structures Calculations, pages 73–107. Progress in Computational Physics, 2010.Google Scholar
  5. 5.
    S. Balay, M. F. Adams, J. Brown, P. Brune, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. Curfman McInnes, K. Rupp, B. F. Smith, and H. Zhang. PETSc Web page. http://www.mcs.anl.gov/petsc, 2015.
  6. 6.
    A. Bayliss, M. Gunzburger, and E. Turkel. Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J. Appl. Math., 42(2):430–451, 1982.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J.-P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 114(2):185–200, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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(2):469–488, 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Y. Boubendir. An analysis of the BEM-FEM non-overlapping domain decomposition method for a scattering problem. J. Comput. Appl. Math., 204(2):282–291, 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Y. Boubendir, X. Antoine, and C. Geuzaine. A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation. Journal of Computational Physics, 231(2):262–280, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Y. Boubendir, A. Bendali, and M. B. Fares. Coupling of a non-overlapping domain decomposition method for a nodal finite element method with a boundary element method. Internat. J. Numer. Methods Engrg., 73(11):1624–1650, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    F. Collino and P. Monk. The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput., 19(6):2061–2090 (electronic), 1998.Google Scholar
  13. 13.
    B. Després. Méthodes de décomposition de domaine pour les problèmes de propagation d’ondes en régime harmonique. Le théorème de Borg pour l’équation de Hill vectorielle. PhD thesis, Rocquencourt, 1991. Thèse, Université de Paris IX (Dauphine), Paris, 1991.Google Scholar
  14. 14.
    B. Després, P. Joly, and J. E. Roberts. A domain decomposition method for the harmonic Maxwell equations. In Iterative methods in linear algebra (Brussels, 1991), pages 475–484, Amsterdam, 1992. North-Holland.Google Scholar
  15. 15.
    V. Dolean, J. M. Gander, S. Lanteri, J.-F. Lee, and Z. Peng. Optimized Schwarz methods for curl-curl time-harmonic Maxwell’s equations. 2013.zbMATHGoogle Scholar
  16. 16.
    V. Dolean, M. J. Gander, and L. Gerardo-Giorda. Optimized Schwarz methods for Maxwell’s equations. SIAM J. Sci. Comput., 31(3):2193–2213, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    P. Dular and C. Geuzaine. GetDP Web page, http:://getdp.info, 2015. [online]. available: http:://getdp.info.
  18. 18.
    P. Dular, C. Geuzaine, F. Henrotte, and W. Legros. A general environment for the treatment of discrete problems and its application to the finite element method. IEEE Transactions on Magnetics, 34(5):3395–3398, September 1998.Google Scholar
  19. 19.
    M. El Bouajaji, X Antoine, and C. Geuzaine. Approximate local magnetic-to-electric surface operators for time-harmonic Maxwell’s equations. Journal of Computational Physics, 279(15):241–260, 2014.Google Scholar
  20. 20.
    M. El Bouajaji, V. Dolean, M. Gander, and S. Lanteri. Optimized Schwarz methods for the time-harmonic Maxwell equations with damping. SIAM Journal on Scientific Computing, 34(4):A2048–A2071, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    M. El Bouajaji, B. Thierry, X. Antoine, and C. Geuzaine. A quasi-optimal domain decomposition algorithm for the time-harmonic Maxwell’s equations. Journal of Computational Physics, 294(1):38–57, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31(139):629–651, 1977.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    B. Engquist and L. Ying. Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers. Multiscale Model. Simul., 9(2):686–710, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    O.G. Ernst and M.J. Gander. Why it is difficult to solve Helmholtz problems with classical iterative methods. In Ivan G. Graham, Thomas Y. Hou, Omar Lakkis, and Robert Scheichl, editors, Numerical Analysis of Multiscale Problems, volume 83 of Lecture Notes in Computational Science and Engineering, pages 325–363. Springer Berlin Heidelberg, 2012.Google Scholar
  25. 25.
    M. Gander. Optimized Schwarz methods. SIAM Journal on Numerical Analysis, 44(2):699–731, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    M. Gander and L. Halpern. Méthode de décomposition de domaine. Encyclopédie électronique pour les ingénieurs, 2012.Google Scholar
  27. 27.
    M. J. Gander, F. Magoulès, and F. Nataf. Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput., 24(1):38–60 (electronic), 2002.Google Scholar
  28. 28.
    C. Geuzaine. GetDP: a general finite-element solver for the de Rham complex. In PAMM Volume 7 Issue 1. Special Issue: Sixth International Congress on Industrial Applied Mathematics (ICIAM07) and GAMM Annual Meeting, Zürich 2007, volume 7, pages 1010603–1010604. Wiley, 2008.Google Scholar
  29. 29.
    C. Geuzaine, F. Henrotte, E. Marchandise, J.-F. Remacle, P. Dular, and R. Vazquez Sabariego. ONELAB: Open Numerical Engineering LABoratory. Proceedings of the 7th European Conference on Numerical Methods in Electromagnetism (NUMELEC2012), 2012.Google Scholar
  30. 30.
    C. Geuzaine, F. Henrotte, E. Marchandise, J.-F. Remacle, and R. Vazquez Sabariego. ONELAB Web page, http:://onelab.info, 2015. [online]. available: http:://onelab.info.
  31. 31.
    C. Geuzaine and J.-F. Remacle. Gmsh Web page, http:://gmsh.info, 2015. [online]. available: http:://gmsh.info.
  32. 32.
    C. Geuzaine and J.-F. Remacle. Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Internat. J. Numer. Methods Engrg., 79(11):1309–1331, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    C. Geuzaine, B. Thierry, N. Marsic, D. Colignon, A. Vion, S. Tournier, Y. Boubendir, M. El Bouajaji, and X. Antoine. An open source domain decomposition solver for time-harmonic electromagnetic wave problems. In Proceedings of the 2014 IEEE Conference on Antenna and Measurements and Applications, CAMA 2014, November 2014.Google Scholar
  34. 34.
    D. Givoli. Computational absorbing boundaries. In Steffen Marburg and Bodo Nolte, editors, Computational Acoustics of Noise Propagation in Fluids - Finite and Boundary Element Methods, pages 145–166. Springer Berlin Heidelberg, 2008.Google Scholar
  35. 35.
    R. Kerchroud, X. Antoine, and A. Soulaimani. Numerical accuracy of a Padé-type non-reflecting boundary condition for the finite element solution of acoustic scattering problems at high-frequency. International Journal for Numerical Methods in Engineering, 64(10):1275–1302, 2005.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    P.-L. Lions. On the Schwarz alternating method. III. A variant for nonoverlapping subdomains. In Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), pages 202–223. SIAM, Philadelphia, PA, 1990.Google Scholar
  37. 37.
    FA Milinazzo, CA Zala, and GH Brooke. Rational square-root approximations for parabolic equation algorithms. Journal of the Acoustical Society of America, 101(2):760–766, FEB 1997.Google Scholar
  38. 38.
    A. Modave, E. Delhez, and C. Geuzaine. Optimizing perfectly matched layers in discrete contexts. International Journal for Numerical Methods in Engineering, 99(6):410–437, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    A. Moiola and E. A. Spence. Is the Helmholtz equation really sign-indefinite? SIAM Rev., 56(2):274–312, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    F. Nataf. Interface connections in domain decomposition methods. NATO Science Series II, 75, 2001.Google Scholar
  41. 41.
    F. Nataf and F. Nier. Convergence rate of some domain decomposition methods for overlapping and nonoverlapping subdomains. Numer. Math., 75:357–377, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    J.-C. Nédélec. Acoustic and electromagnetic equations, volume 144 of Applied Mathematical Sciences. Springer-Verlag, New York, 2001. Integral representations for harmonic problems.Google Scholar
  43. 43.
    Z. Peng and J. Lee. A scalable nonoverlapping and nonconformal domain decomposition method for solving time-harmonic Maxwell equations in \(\mathbb{R}^{3}\). SIAM Journal on Scientific Computing, 34(3):A1266–A1295, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Z. Peng, V. Rawat, and J.-F. Lee. One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems. Journal of Computational Physics, 229(4):1181–1197, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    V. Rawat and J.-F. Lee. Nonoverlapping domain decomposition with second order transmission condition for the time-harmonic Maxwell’s equations. SIAM J. Scientific Computing, 32(6):3584–3603, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Y. Saad and M. H. Schultz. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 7(3):856–869, 1986.MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    C. Stolk. A rapidly converging domain decomposition method for the Helmholtz equation. Journal of Computational Physics, 241(0):240–252, 2013.Google Scholar
  48. 48.
    B. Thierry, A.Vion, S. Tournier, M. El Bouajaji, D. Colignon, N. Marsic, X. Antoine, and C. Geuzaine. GetDDM: an open framework for testing optimized Schwarz methods for time-harmonic wave problems. Submitted to Computer Physics Communications, 2015.Google Scholar
  49. 49.
    A. Toselli and O. Widlund. Domain decomposition methods—algorithms and theory, volume 34 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2005.Google Scholar
  50. 50.
    A. Vion, R. Bélanger-Rioux, L. Demanet, and C. Geuzaine. A DDM double sweep preconditioner for the Helmholtz equation with matrix probing of the DtN map. In Mathematical and Numerical Aspects of Wave Propagation WAVES 2013, June 2013.Google Scholar
  51. 51.
    A. Vion and C. Geuzaine. Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem. Journal of Computational Physics, 266(0):171–190, 2014.Google Scholar
  52. 52.
    A. Vion and C. Geuzaine. Parallel double sweep preconditioner for the optimized Schwarz algorithm applied to high frequency Helmholtz and Maxwell equations. In LNCSE, Proc. of DD22, 2014.Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut Elie Cartan de LorraineUniversité de Lorraine, Inria Nancy-Grand Est EPI SPHINXVandoeuvre-lès-Nancy CedexFrance
  2. 2.Institut Montefiore B28Université de LiègeLiègeBelgium

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