Inverted finite elements : a new method for solving elliptic problems in unbounded domains

Boulmezaoud, Tahar Zamène

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005), p. 109-145 / Harvested from Numdam

Résumé

In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of $\mathbb {R}^n$. The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used for describing the growth or the decay of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate its efficiency and its high performance.

Détail

Publié le : 2005-01-01

MR 2136202 | Zbl 1078.65102

DOI : https://doi.org/10.1051/m2an:2005001
Classification:  35J,  35J05,  65Jxx,  65Nxx,  65Rxx

Bibliographie

[1] R.A. Adams, Compact imbeddings of weighted Sobolev spaces on unbounded domains. J. Differential Equations 9 (1971) 325-334. | Zbl 0209.15004

[2] F. Alliot and C. Amrouche, Problème de Stokes dans $\mathbb {R}^n$ et espaces de Sobolev avec poids. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 1247-1252. | Zbl 0894.35082

[3] C. Amrouche, V. Girault and J. Giroire, Weighted Sobolev spaces for Laplace’s equation in $\mathbb {R}^n$. J. Math. Pures Appl. (9) 73 (1994) 579-606. | Zbl 0836.35038

[4] C. Amrouche, V. Girault and J. Giroire, Dirichlet and Neumann exterior problems for the $n$-dimensional Laplace operator: an approach in weighted Sobolev spaces. J. Math. Pures Appl. (9) 76 (1997) 55-81. | Zbl 0878.35029

[5] J. Bérenger, A perfectly matched layer for absoption of electromagnetics waves. J. Comput. Physics 114 (1994) 185-200. | Zbl 0814.65129

[6] J. Bérenger, Perfectly matched layer for the fdtd solution of wave-structure interaction problems. IEEE Trans. Antennas Propagat. 44 (1996) 110-117.

[7] P. Bettess and O.C. Zienkiewicz, Diffraction and refraction of surface waves using finite and infinite elements. Internat. J. Numer. Methods Engrg. 11 (1977) 1271-1290. | Zbl 0367.76014

[8] T.Z. Boulmezaoud, Vector potentials in the half-space of $\mathbb {R}^3$. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 711-716. | Zbl 0981.35013

[9] T.Z. Boulmezaoud, On the Stokes system and on the biharmonic equation in the half-space: an approach via weighted Sobolev spaces. Math. Methods Appl. Sci. 25 (2002) 373-398. | Zbl 0998.35037

[10] T.Z. Boulmezaoud, On the Laplace operator and on the vector potential problems in the half-space: an approach using weighted spaces. Math. Methods Appl. Sci. 26 (2003) 633-669. | Zbl pre01925341

[11] D.S. Burnett, A three-dimensional acoustic infinite element based on a prolate spheroidal multipole expansion. J. Acoust. Soc. Amer. 96 (1994) 2798-2816.

[12] C. Canuto, S.I. Hariharan, L. Lustman, Spectral methods for exterior elliptic problems. Numer. Math. 46 (1985) 505-520. | Zbl 0548.65082

[13] Y. Choquet-Bruhat and D. Christodoulou, Elliptic systems in $H_{s,\delta }$ spaces on manifolds which are Euclidean at infinity. Acta Math. 146 (1981) 129-150. | Zbl 0484.58028

[14] Ph.-G. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[15] D.L. Colton and R. Kress, Integral equation methods in scattering theory. Pure Appl. Math. John Wiley & Sons Inc., New York (1983). | MR 700400 | Zbl 0522.35001

[16] L. Demkowicz and F. Ihlenburg, Analysis of a coupled finite-infinite element method for exterior Helmholtz problems. Numer. Math. 88 (2001) 43-73. | Zbl 1013.65122

[17] J. Deny and J.L. Lions, Les espaces du type de Beppo Levi. Ann. Inst. Fourier, Grenoble 5 (1955) 305-370, (1953-54). | Numdam | Zbl 0065.09903

[18] K. Gerdes, A summary of infinite element formulations for exterior Helmholtz problems. Comput. Methods Appl. Mech. Engrg. 164 (1998) 95-105. | Zbl 0943.65126

[19] K. Gerdes and L. Demkowicz, Solution of $3$D-Laplace and Helmholtz equations in exterior domains using $hp$-infinite elements. Comput. Methods Appl. Mech. Engrg. 137 (1996) 239-273. | Zbl 0881.73126

[20] V. Girault, The divergence, curl and Stokes operators in exterior domains of $\mathbb {R}^n$. In Recent developments in theoretical fluid mechanics (Paseky, 1992), Pitman Res. Notes Math. Ser. Longman Sci. Tech., Harlow 291 (1993) 34-77. | Zbl 0837.35115

[21] V. Girault, The Stokes problem and vector potential operator in three-dimensional exterior domains: an approach in weighted Sobolev spaces. Differential Integral Equations 7 (1994) 535-570. | Zbl 0831.35125

[22] J. Giroire, Étude de quelques problèmes aux limites extérieures et résolution par équations intégrales. Thèse de Doctorat d'Etat, Université Pierre et Marie Curie, Paris (1987).

[23] J. Giroire and J.-C. Nédélec, Numerical solution of an exterior Neumann problem using a double layer potential. Math. Comp. 32 (1978) 973-990. | Zbl 0405.65060

[24] L. Halpern, A spectral method for the Stokes problem in three-dimensional unbounded domains. Math. Comp. 70 (2001) 1417-1436 (electronic). | Zbl 0971.35006

[25] B. Hanouzet, Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace. Rend. Sem. Mat. Univ. Padova 46 (1971) 227-272. | Numdam | Zbl 0247.35041

[26] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). | JFM 60.0169.01 | MR 944909 | Zbl 0634.26008

[27] L. Hörmander and J.L. Lions, Sur la complétion par rapport à une intégrale de Dirichlet. Math. Scand. 4 (1956) 259-270. | Zbl 0078.28003

[28] F. Ihlenburg, Finite element analysis of acoustic scattering, volume 132 of Applied Mathematical Sciences. Springer-Verlag, New York (1998). | MR 1639879 | Zbl 0908.65091

[29] V.A. Kondratev, Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč. 16 (1967) 209-292. | Zbl 0194.13405

[30] A. Kufner, Weighted Sobolev spaces. A Wiley-Interscience Publication. John Wiley & Sons Inc., New York (1985). | MR 802206 | Zbl 0567.46009

[31] M. Laib and T.Z. Boulmezaoud, Some properties of weighted sobolev spaces in unbounded domains. In preparation.

[32] M.N. Le Roux, Méthode d’éléments finis pour la résolution numérique de problèmes extérieurs en dimension $2$. RAIRO Anal. Numér. 11 (1977) 27-60. | Numdam | Zbl 0382.65055

[33] V.G. Maz'Ya and B.A. Plamenevskii, Weighted spaces with inhomogeneous norms, and boundary value problems in domains with conical points, in Elliptische Differentialgleichungen (Meeting, Rostock, 1977). Wilhelm-Pieck-Univ. Rostock (1978) 161-190. | Zbl 0429.35031

[34] J.-C. Nédélec, Curved finite element methods for the solution of singular integral equations on surfaces in $\mathbb {R}^3$. Comput. Methods Appl. Mech. Engrg. 8 (1976) 61-80. | Zbl 0333.45015

[35] J.-C. Nédélec. Résolution des Équations de Maxwell par Méthodes Intégrales. Cours de D.E.A. École Polytechnique, Paris (1998).

[36] V. Rokhlin, Solution of acoustic scattering problems by means of second kind integral equations. Wave Motion 5 (1983) 257-272. | Zbl 0522.73022