Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-09T21:34:51.791Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 March 2005

Tahar Zamène Boulmezaoud
Tahar Zamène Boulmezaoud*
Affiliation:
Laboratoire de Mathématiques, Université de Versailles Saint-Quentin en Yvelines (UVSQ), 45 avenue des États-Unis, Bâtiment Fermat, 78035 Versailles, France. Laboratoire Jacques-Louis Lions, Université Paris VI, BC187, 75252 Paris Cedex, France. boulmeza@math.uvsq.fr
Get access

Abstract

In this paper, we propose a new numerical method for solvingelliptic equations in unbounded regions of ${\mathbb{R}}^n$ . Themethod is based on the mapping of a part of the domain into abounded region. An appropriate family of weighted spaces is usedfor describing the growth or the decay of functions at large distances. Afterexposing the main ideas of the method, we analysecarefully its convergence. Some 3D computational results are displayedto demonstrate its efficiency and its high performance.

Keywords

Unbounded domainsinverted elements methodweightedSobolev spaces.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 1 , January 2005 , pp. 109 - 145
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Adams, R.A., Compact imbeddings of weighted Sobolev spaces on unbounded domains. J. Differential Equations 9 (1971) 325–334. CrossRef
Alliot, F. and Amrouche, C., 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. CrossRef
Amrouche, C., Girault, V. and Giroire, J., Weighted Sobolev spaces for Laplace's equation in ${\mathbb{R}}^n$ . J. Math. Pures Appl. (9) 73 (1994) 579–606.
Amrouche, C., Girault, V. and Giroire, J., 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. CrossRef
Bérenger, J., A perfectly matched layer for absoption of electromagnetics waves. J. Comput. Physics 114 (1994) 185–200.
Bérenger, J., Perfectly matched layer for the fdtd solution of wave-structure interaction problems. IEEE Trans. Antennas Propagat. 44 (1996) 110–117.
Bettess, P. and Zienkiewicz, O.C., Diffraction and refraction of surface waves using finite and infinite elements. Internat. J. Numer. Methods Engrg. 11 (1977) 1271–1290. CrossRef
Boulmezaoud, T.Z., Vector potentials in the half-space of $\Bbb R\sp 3$ . C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 711–716. CrossRef
Boulmezaoud, T.Z., 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. CrossRef
Boulmezaoud, T.Z., 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. CrossRef
Burnett, D.S., A three-dimensional acoustic infinite element based on a prolate spheroidal multipole expansion. J. Acoust. Soc. Amer. 96 (1994) 2798–2816. CrossRef
Canuto, C., Hariharan, S.I., Lustman, L., Spectral methods for exterior elliptic problems. Numer. Math. 46 (1985) 505–520. CrossRef
Choquet-Bruhat, Y. and Christodoulou, D., Elliptic systems in $H\sb{s,\delta }$ spaces on manifolds which are Euclidean at infinity. Acta Math. 146 (1981) 129–150. CrossRef
Ph.-G. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam (1978).
D.L. Colton and R. Kress, Integral equation methods in scattering theory. Pure Appl. Math. John Wiley & Sons Inc., New York (1983).
Demkowicz, L. and Ihlenburg, F., Analysis of a coupled finite-infinite element method for exterior Helmholtz problems. Numer. Math. 88 (2001) 43–73. CrossRef
J. Deny and J.L. Lions, Les espaces du type de Beppo Levi. Ann. Inst. Fourier, Grenoble 5 (1955) 305–370, (1953–54).
Gerdes, K., A summary of infinite element formulations for exterior Helmholtz problems. Comput. Methods Appl. Mech. Engrg. 164 (1998) 95–105. CrossRef
Gerdes, K. and Demkowicz, L., Solution of 3D-Laplace and Helmholtz equations in exterior domains using hp-infinite elements. Comput. Methods Appl. Mech. Engrg. 137 (1996) 239–273. CrossRef
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.
Girault, V., 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.
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).
Giroire, J. and Nédélec, J.-C., Numerical solution of an exterior Neumann problem using a double layer potential. Math. Comp. 32 (1978) 973–990. CrossRef
Halpern, L., A spectral method for the Stokes problem in three-dimensional unbounded domains. Math. Comp. 70 (2001) 1417–1436 (electronic). CrossRef
Hanouzet, B., Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace. Rend. Sem. Mat. Univ. Padova 46 (1971) 227–272.
G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988).
Hörmander, L. and Lions, J.L., Sur la complétion par rapport à une intégrale de Dirichlet. Math. Scand. 4 (1956) 259–270. CrossRef
F. Ihlenburg, Finite element analysis of acoustic scattering, volume 132 of Applied Mathematical Sciences. Springer-Verlag, New York (1998).
V.A. Kondratev, Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Ob v s v c. 16 (1967) 209–292.
A. Kufner, Weighted Sobolev spaces. A Wiley-Interscience Publication. John Wiley & Sons Inc., New York (1985).
M. Laib and T.Z. Boulmezaoud, Some properties of weighted sobolev spaces in unbounded domains. In preparation.
Le Roux, M.N., Méthode d'éléments finis pour la résolution numérique de problèmes extérieurs en dimension 2. RAIRO Anal. Numér. 11 (1977) 27–60. CrossRef
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.
Nédélec, J.-C., 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.
J.-C. Nédélec. Résolution des Équations de Maxwell par Méthodes Intégrales. Cours de D.E.A. École Polytechnique, Paris (1998).
Rokhlin, V., Solution of acoustic scattering problems by means of second kind integral equations. Wave Motion 5 (1983) 257–272. CrossRef