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Partition of unity method for Helmholtz equation: q-convergence for plane-wave and wave-band local bases

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Abstract

In this paper we study the q-version of the Partition of Unity Method for the Helmholtz equation. The method is obtained by employing the standard bilinear finite element basis on a mesh of quadrilaterals discretizing the domain as the Partition of Unity used to paste together local bases of special wave-functions employed at the mesh vertices. The main topic of the paper is the comparison of the performance of the method for two choices of local basis functions, namely a) plane-waves, and b) wave-bands. We establish the q-convergence of the method for the class of analytical solutions, with q denoting the number of plane-waves or wave-bands employed at each vertex, for which we get better than exponential convergence for sufficiently small h, the mesh-size of the employed mesh. We also discuss the a-posteriori estimation for any solution quantity of interest and the problem of quadrature for all integrals employed. The goal of the paper is to stimulate theoretical development which could explain various numerical features. A main open question is the analysis of the pollution and its disappearance as function of h and q.

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References

  1. I. Babuška, G. Caloz, and J. E. Osborn: Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31 (1994), 945–981.

    Article  MathSciNet  Google Scholar 

  2. J. M. Melenk, I. Babuška: Approximation with harmonic and generalized harmonic polynomials in the partition of unity method. Comput. Assist. Mech. Eng. Sci. 4 (1997), 607–632.

    Google Scholar 

  3. J. M. Melenk, I. Babuška: The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Eng. 139_(1996), 289–314.

    Article  Google Scholar 

  4. I. Babuška, J. M. Melenk: The partition of unity method. Int. J. Numer. Methods. Eng. 40 (1997), 727–758.

    Article  Google Scholar 

  5. J. M. Melenk: On generalized finite element methods. PhD. thesis. University of Maryland, 1995.

  6. I. Babuška, T. Strouboulis, and K. Copps: The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Eng. 181 (2000), 43–69.

    Article  Google Scholar 

  7. T. Strouboulis, I. Babuška, and K. Copps: The generalized finite element method: An example of its implementation and illustration of its performance. Int. J. Numer. Methods Eng. 47 (2000), 1401–1417.

    Article  Google Scholar 

  8. T. Strouboulis, K. Copps, and I. Babuška: The generalized finite element method. Comput. Methods Appl. Mech. Eng. 190 (2001), 4081–4193.

    Article  Google Scholar 

  9. T. Strouboulis, L. Zhang, and I. Babuška: Generalized finite element method using mesh-based handbooks: Application to problem in domains with many voids. Comput. Methods Appl. Mech. Eng. 192 (2003), 3109–3161.

    Article  Google Scholar 

  10. T. Strouboulis, L. Zhang, and I. Babuška: p-version of the generalized FEM using mesh-based handbooks with applications to multiscale problems. Int. J. Numer. Methods Eng. 60 (2004), 1639–1672.

    Article  Google Scholar 

  11. N. Sukumar, D. L. Chopp, N. Moës, and T. Belytschko: Modeling holes and inclusions by level sets in the extended finite element method. Comput. Methods Appl. Mech. Eng. 190 (2001), 6183–6200.

    Article  Google Scholar 

  12. O. Laghrouche, P. Bettess: Solving short wave problems using special finite elements towards an adaptive approach. In: The Mathematics of Finite Elements and Applications X (J. R. Whiteman, ed.). Elsevier, Amsterdam, 1999, pp. 181–194.

    Google Scholar 

  13. P. Bettess, J. Shirron, O. Laghrouche, B. Peseux, R. Sugimoto, and J. Trevelyan: A numerical integration scheme for special finite elements for the Helmholtz equation. Int. J. Numer. Methods Eng. 56 (2003), 531–552.

    Article  Google Scholar 

  14. R. Sugimoto, P. Bettess, and J. Trevelyan: A numerical integration scheme for special quadrilateral finite elements for the Helmholtz equation. Commun. Numer. Methods Eng. 19 (2003), 233–245.

    Article  MathSciNet  Google Scholar 

  15. E. Perrey-Debain, O. Laghrouche, P. Bettess, and J. Trevelyan: Plane-wave basis finite elements and boundary elements for three-dimensional wave scattering. Philos. Trans. R. Soc. Lond. A 362 (2004), 561–577.

    Article  MathSciNet  Google Scholar 

  16. O. Laghrouche, P. Bettess, E. Perrey-Debain, and J. Trevelyan: Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed. Comput. Methods Appl. Mech. Eng. 194 (2005), 367–381.

    Article  MathSciNet  Google Scholar 

  17. P. Ortiz, E. Sanchez: An improved partition of unity finite element model for diffraction problems. Int. J. Numer. Methods Eng. 50 (2001), 2727–2740.

    Article  Google Scholar 

  18. P. Ortiz: Finite elements using a plane-wave basis for scattering of surface water waves. Philos. Trans. R. Soc. Lond. A 362 (2004), 525–540.

    Article  MATH  MathSciNet  Google Scholar 

  19. C. Farhat, I. Harari, and L.P. Franca: The discontinuous enrichment method. Comput. Methods Appl. Mech. Eng. 190 (2001), 6455–6479.

    Article  MathSciNet  Google Scholar 

  20. C. Farhat, I. Harari, and U. Hetmaniuk: A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Eng. 192 (2003), 1389–1419.

    Article  MathSciNet  Google Scholar 

  21. P. Rouch, P. Ladev-eze: The variational theory of complex rays: A predictive tool for medium-frequency vibrations. Comput. Methods Appl. Mech. Eng. 192 (2003), 3301–3315.

    Article  Google Scholar 

  22. H. Riou, P. Ladevèze, and P. Rouch: Extension of the variational theory of complex rays to shells for medium-freqency vibrations. J. Sound Vib. 272 (2004), 341–360.

    Article  Google Scholar 

  23. P. Ladevèze, H. Riou: Calculation of medium-frequency vibrations over a wide frequency range. Comput. Methods Appl. Mech. Eng. 194 (2005), 3167–3191.

    Article  Google Scholar 

  24. I. Babuška, F. Ihlenburg, T. Strouboulis, and S. K. Gangaraj: A posteriori error estimation for finite element solutions of Helmholtz’ equation. Part I: The quality of local indicators and estimators. Int. J. Numer. Methods Eng. 40_(1997), 3443–3462.

    Article  Google Scholar 

  25. I. Babuška, F. Ihlenburg, T. Strouboulis, and S. K. Gangaraj: A posteriori error estimation for finite element solutions of Helmholtz’ equation. Part II: Estimation of the pollution error. Int. J. Numer. Methods Eng. 40 (1997), 3883–3900.

    Article  Google Scholar 

  26. F. Ihlenburg: Finite Element Analysis of Acoustic Scattering. Springer-Verlag, New York, 1998.

    Google Scholar 

  27. I. Babuška, S. Sauter: Is the pollution e-ect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal. 34 (1997), 2392–2423.

    MathSciNet  Google Scholar 

  28. T. Strouboulis, I. Babuška, and R. Hidajat: The generalized finite element method for Helmholtz equation: theory, computation, and open problems. Comput. Methods Appl. Mech. Eng. Accepted for publication.

  29. P. J. Davis, P. Rabinowitz: Methods of Numerical Integration. Academic Press, San Diego, 1984.

    Google Scholar 

  30. D. S. Jones: Acoustic and Electromagnetic Waves. Oxford University Press, New York, 1986.

    Google Scholar 

  31. B. A. Szabó, I. Babuška: Finite Element Analysis. John Wiley & Sons, New York, 1991.

    Google Scholar 

  32. T. Strouboulis, L. Zhang, D. Wang, and I. Babuška: A posteriori error estimation for generalized finite element methods. Comput. Methods Appl. Mech. Eng. 195 (2006), 852–879.

    Article  Google Scholar 

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Authors and Affiliations

  1. Dept. of Aerospace Engineering, Texas A&M University, College Station, TX, 77843, USA

    Theofanis Strouboulis & Realino Hidajat

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  1. Theofanis Strouboulis
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  2. Realino Hidajat
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Additional information

This work was supported by the Office of Naval Research under Grant N00014-99-1-0726. The support of Dr. Luise Couchman of the Office of Naval Research is greatly appreciated.

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Strouboulis, T., Hidajat, R. Partition of unity method for Helmholtz equation: q-convergence for plane-wave and wave-band local bases. Appl Math 51, 181–204 (2006). https://doi.org/10.1007/s10492-006-0011-0

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