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

 Access to full text

 Full (PDF)

1. [1] R.A. Adams, Compact imbeddings of weighted Sobolev spaces on unbounded domains. J. Differential Equations 9 (1971) 325–334. Zbl0209.15004
2. [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. Zbl0894.35082
3. [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. Zbl0836.35038
4. [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. Zbl0878.35029
5. [5] J. Bérenger, A perfectly matched layer for absoption of electromagnetics waves. J. Comput. Physics 114 (1994) 185–200. Zbl0814.65129
6. [6] J. Bérenger, Perfectly matched layer for the fdtd solution of wave-structure interaction problems. IEEE Trans. Antennas Propagat. 44 (1996) 110–117. 
7. [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. Zbl0367.76014
8. [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. Zbl0981.35013
9. [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. Zbl0998.35037
10. [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. Zbl1290.35034
11. [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. [12] C. Canuto, S.I. Hariharan, L. Lustman, Spectral methods for exterior elliptic problems. Numer. Math. 46 (1985) 505–520. Zbl0548.65082
13. [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. Zbl0484.58028
14. [14] Ph.-G. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam (1978). Zbl0383.65058MR520174
15. [15] D.L. Colton and R. Kress, Integral equation methods in scattering theory. Pure Appl. Math. John Wiley & Sons Inc., New York (1983). Zbl0522.35001MR700400
16. [16] L. Demkowicz and F. Ihlenburg, Analysis of a coupled finite-infinite element method for exterior Helmholtz problems. Numer. Math. 88 (2001) 43–73. Zbl1013.65122
17. [17] J. Deny and J.L. Lions, Les espaces du type de Beppo Levi. Ann. Inst. Fourier, Grenoble 5 (1955) 305–370, (1953–54). Zbl0065.09903
18. [18] K. Gerdes, A summary of infinite element formulations for exterior Helmholtz problems. Comput. Methods Appl. Mech. Engrg. 164 (1998) 95–105. Zbl0943.65126
19. [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. Zbl0881.73126
20. [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. Zbl0837.35115
21. [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. Zbl0831.35125
22. [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. [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. Zbl0405.65060
24. [24] L. Halpern, A spectral method for the Stokes problem in three-dimensional unbounded domains. Math. Comp. 70 (2001) 1417–1436 (electronic). Zbl0971.35006
25. [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. Zbl0247.35041
26. [26] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). Zbl0634.26008MR944909JFM60.0169.01
27. [27] L. Hörmander and J.L. Lions, Sur la complétion par rapport à une intégrale de Dirichlet. Math. Scand. 4 (1956) 259–270. Zbl0078.28003
28. [28] F. Ihlenburg, Finite element analysis of acoustic scattering, volume 132 of Applied Mathematical Sciences. Springer-Verlag, New York (1998). Zbl0908.65091MR1639879
29. [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. Zbl0194.13405
30. [30] A. Kufner, Weighted Sobolev spaces. A Wiley-Interscience Publication. John Wiley & Sons Inc., New York (1985). Zbl0567.46009MR802206
31. [31] M. Laib and T.Z. Boulmezaoud, Some properties of weighted sobolev spaces in unbounded domains. In preparation. 
32. [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. Zbl0382.65055
33. [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. Zbl0429.35031
34. [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. Zbl0333.45015
35. [35] J.-C. Nédélec. Résolution des Équations de Maxwell par Méthodes Intégrales. Cours de D.E.A. École Polytechnique, Paris (1998). 
36. [36] V. Rokhlin, Solution of acoustic scattering problems by means of second kind integral equations. Wave Motion 5 (1983) 257–272. Zbl0522.73022