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

Boulmezaoud, Tahar Zamène

ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 1, pp. 109-145.

In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of ${ℝ}^n$. 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.

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

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

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

[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

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

[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

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

[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

[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.

[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

[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

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

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

[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

[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

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

[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

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

[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

[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

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

[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

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

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

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

[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

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

[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

[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

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

[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