Previous |  Up |  Next

Article

Keywords:
Partition of Unity Method (PUM); Helmholtz equation; exponential convergence; extrapolation; pollution due to wave-number
Summary:
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 $q$.
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. DOI 10.1137/0731051 | MR 1286212
[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.
[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. DOI 10.1016/S0045-7825(96)01087-0 | MR 1426012
[4] I.  Babuška, J. M.  Melenk: The partition of unity method. Int. J. Numer. Methods. Eng. 40 (1997), 727–758. MR 1429534
[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. DOI 10.1016/S0045-7825(99)00072-9 | MR 1734667
[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. DOI 10.1002/(SICI)1097-0207(20000320)47:8<1401::AID-NME835>3.0.CO;2-8 | MR 1746489
[8] T.  Strouboulis, K.  Copps, and I.  Babuška: The generalized finite element method. Comput. Methods Appl. Mech. Eng. 190 (2001), 4081–4193. DOI 10.1016/S0045-7825(01)00188-8 | MR 1832655
[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. DOI 10.1016/S0045-7825(03)00347-5 | MR 2007029
[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. DOI 10.1002/nme.1017 | MR 2069141
[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. DOI 10.1016/S0045-7825(01)00215-8 | MR 1857695
[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. MR 1801975
[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. DOI 10.1002/nme.575
[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. DOI 10.1002/cnm.584 | MR 1965603
[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. DOI 10.1098/rsta.2003.1335 | MR 2075907
[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. DOI 10.1016/j.cma.2003.12.074 | MR 2105168
[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. DOI 10.1002/nme.161
[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. DOI 10.1098/rsta.2003.1333 | MR 2075905 | Zbl 1083.76045
[19] C.  Farhat, I.  Harari, and L. P.  Franca: The discontinuous enrichment method. Comput. Methods Appl. Mech. Eng. 190 (2001), 6455–6479. DOI 10.1016/S0045-7825(01)00232-8 | MR 1870426
[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. DOI 10.1016/S0045-7825(02)00646-1 | MR 1963058
[21] P.  Rouch, P.  Ladevèze: The variational theory of complex rays: A predictive tool for medium-frequency vibrations. Comput. Methods Appl. Mech. Eng. 192 (2003), 3301–3315. DOI 10.1016/S0045-7825(03)00352-9
[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. DOI 10.1016/S0022-460X(03)00775-2
[23] P.  Ladevèze, H.  Riou: Calculation of medium-frequency vibrations over a wide frequency range. Comput. Methods Appl. Mech. Eng. 194 (2005), 3167–3191. DOI 10.1016/j.cma.2004.08.009 | MR 2142539
[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. DOI 10.1002/(SICI)1097-0207(19970930)40:18<3443::AID-NME221>3.0.CO;2-1 | MR 1471617
[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. DOI 10.1002/(SICI)1097-0207(19971115)40:21<3883::AID-NME231>3.0.CO;2-V | MR 1475346
[26] F. Ihlenburg: Finite Element Analysis of Acoustic Scattering. Springer-Verlag, New York, 1998. MR 1639879 | Zbl 0908.65091
[27] I.  Babuška, S.  Sauter: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM J.  Numer. Anal. 34 (1997), 2392–2423. MR 1480387
[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. MR 2240576
[29] P. J.  Davis, P. Rabinowitz: Methods of Numerical Integration. Academic Press, San Diego, 1984. MR 0760629
[30] D. S.  Jones: Acoustic and Electromagnetic Waves. Oxford University Press, New York, 1986. MR 0943347
[31] B.  A.  Szabó, I.  Babuška: Finite Element Analysis. John Wiley & Sons, New York, 1991. MR 1164869
[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. DOI 10.1016/j.cma.2005.03.004 | MR 2195292
Partner of
EuDML logo