We propose a new spectral method for solving multi-dimensional second order elliptic equations with varying coefficients in the whole space. This method employs an orthogonal family of quasi-rational functions recently discovered by Arar and Boulmezaoud. After proving an error estimate, we present some computational tests which demonstrate the efficiency of the method and the significance of its developmental potential.
Mots-clés : Unbounded domains, spectral methods, rational functions, approximation, the whole space
@article{M2AN_2016__50_1_263_0, author = {Boulmezaoud, T.Z. and Arar, N. and Kerdid, N. and Kourta, A.}, title = {Discretization by rational and quasi-rational functions of multi-dimensional elliptic problems in the whole space}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {263--288}, publisher = {EDP-Sciences}, volume = {50}, number = {1}, year = {2016}, doi = {10.1051/m2an/2015042}, zbl = {1337.65164}, mrnumber = {3460109}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015042/} }
TY - JOUR AU - Boulmezaoud, T.Z. AU - Arar, N. AU - Kerdid, N. AU - Kourta, A. TI - Discretization by rational and quasi-rational functions of multi-dimensional elliptic problems in the whole space JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 263 EP - 288 VL - 50 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015042/ DO - 10.1051/m2an/2015042 LA - en ID - M2AN_2016__50_1_263_0 ER -
%0 Journal Article %A Boulmezaoud, T.Z. %A Arar, N. %A Kerdid, N. %A Kourta, A. %T Discretization by rational and quasi-rational functions of multi-dimensional elliptic problems in the whole space %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 263-288 %V 50 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015042/ %R 10.1051/m2an/2015042 %G en %F M2AN_2016__50_1_263_0
Boulmezaoud, T.Z.; Arar, N.; Kerdid, N.; Kourta, A. Discretization by rational and quasi-rational functions of multi-dimensional elliptic problems in the whole space. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 1, pp. 263-288. doi : 10.1051/m2an/2015042. http://www.numdam.org/articles/10.1051/m2an/2015042/
Rotationally invariant quadratures for the sphere. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 465 (2009) 3103–3125. | MR | Zbl
and ,Weighted Sobolev spaces for Laplace’s equation in . J. Math. Pures Appl. 73 (1994) 579–606. | MR | Zbl
, and ,Eigenfunctions of a weighted Laplace operator in the whole space. J. Math. Anal. Appl. 400 (2013) 161–173. | DOI | MR | Zbl
and ,K. Atkinson and W. Han, Spherical harmonics and approximations on the unit sphere: An introduction. Vol. 2044 of Lect. Notes Math. Springer, Heidelberg (2012). | MR | Zbl
Radiation boundary conditions for wavelike equations. Commun. Pure Appl. Math. 33 (1980) 707–725. | DOI | MR | Zbl
and ,A perfectly matched layer for absorption of electromagnetics waves. J. Comput. Phys. 114 (1994) 185–200. | DOI | MR | Zbl
,Perfectly matched layer for the fdtd solution of wave-structure interaction problems. IEEE Trans. Antennas Propag. 44 (1996) 110–117,. | DOI
.Ch. Bernardi and Y. Maday, Approximations spectrales de problèmes aux limites elliptiques, Vol. 10 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer–Verlag, Paris (1992). | MR | Zbl
Infinite elements. Int. J. Numer. Methods Engrg. 11 (1977) 53–64. | DOI | Zbl
,Diffraction and refraction of surface waves using finite and infinite elements. Int. J. Numer. Methods Engrg. 11 (1977) 1271–1290. | DOI | MR | Zbl
and .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. | DOI | MR | Zbl
.On the invariance of weighted Sobolev spaces under Fourier transform. C. R. Math. Acad. Sci. Paris 339 (2004) 861–866. | DOI | MR | Zbl
,Inverted finite elements: a new method for solving elliptic problems in unbounded domains. ESAIM: M2AN 39 (2005) 109–145. | DOI | Numdam | MR | Zbl
,T.Z. Boulmezaoud and K. Kaliche, A new numerical method for the model of solvation in continuum anisotropic dielectrics. In preparation (2015).
On the steady Oseen problem in the whole space. Hiroshima Math. J. 35 (2005) 371–401. | DOI | MR | Zbl
and ,Numerical approximation of second-order elliptic problems in unbounded domains. J. Sci. Comput. 60 (2014) 295–312. | DOI | MR | Zbl
, and ,Orthogonal rational functions on a semi-infinite interval. J. Comput. Phys. 70 (1987) 63–88. | DOI | MR | Zbl
,Spectral methods using rational basis functions on an infinite interval. J. Comput. Phys. 69 (1987) 112–142. | DOI | MR | Zbl
,C.A. Brebbia, J.C.F. Telles and L.C. Wrobel, Boundary Element Techniques. Springer-Verlag, Berlin (1984). | MR | Zbl
F. Brezzi, C. Johnson and J.-C. Nédélec, On the Coupling of Boundary Integral and Finite Element Methods. In Proc. of the Fourth Symposium on Basic Problems of Numerical Mathematics Plzevn. Charles Univ., Prague (1978) 103–114. | MR | Zbl
A three-dimensional acoustic infinite element based on a prolate spheroidal multipole expansion. J. Acoust. Soc. Amer. 96 (1994) 2798–2816. | DOI | MR
,Spectral methods for exterior elliptic problems. Numer. Math. 46 (1985) 505–520. | DOI | MR | Zbl
, and ,On the -adaptive coupling of FE and BE for viscoplastic and elastoplastic interface problems. J. Comput. Appl. Math. 75 (1996) 345–363. | DOI | MR | Zbl
, and ,Approximation variationnelle des problèmes aux limites. Ann. Inst. Fourier Grenoble 14 (1964) 345–444. | DOI | Numdam | MR | Zbl
,Ph.-G. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam (1978). | MR | Zbl
D.L. Colton and R. Kress, Integral equation methods in scattering theory. Pure Appl. Math. John Wiley & Sons Inc., New York (1983). | MR | Zbl
R. Cools, Constructing cubature formulae: the science behind the art. In vol. 6 of Acta Numer. Cambridge Univ. Press, Cambridge (1997) 1–54. | MR | Zbl
Coupling of finite and boundary element methods for an elastoplastic interface problem. SIAM J. Numer. Anal. 27 (1990) 1212–1226. | DOI | MR | Zbl
and ,Symmetric coupling of finite elements and boundary elements for a parabolic-elliptic interface problem. Quart. Appl. Math. 48 (1990) 265–279. | DOI | MR | Zbl
, and ,Absorbing boundary conditions for the numerical simulation of waves. Math. Comput. 31 (1977) 629–651. | DOI | MR | Zbl
and ,Radiation boundary conditions for acoustic and elastic wave calculations. Commun. Pure Appl. Math. 32 (1979) 313–357. | DOI | MR | Zbl
and ,Computational aspects of pseudospectral Laguerre approximations. Appl. Numer. Math. 6 (1990) 447–457. | DOI | MR | Zbl
,Approximations of some diffusion evolution equations in unbounded domains by Hermite functions. Math. Comput. 57 (1990) 597–619. | DOI | MR | Zbl
and ,Solution of D-Laplace and Helmholtz equations in exterior domains using -infinite elements. Comput. Methods Appl. Mech. Engrg. 137 (1996) 239–273. | DOI | MR | Zbl
and ,J. Giroire, Études de quelques problèmes aux limites extérieurs et résolution par équations intégrales. Ph.D. thesis, Université Pierre et Marie Curie, Paris (1987).
Numerical solution of an exterior Neumann problem using a double layer potential. Math. Comput. 32 (1978) 973–990. | DOI | MR | Zbl
and ,A rational approximation and its applications to differential equations on the half line. J. Sci. Comput. 15 (2000) 117–147. | DOI | MR | Zbl
, and ,Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace. Rend. Sem. Mat. Univ. Padova 46 (1971) 227–272. | Numdam | MR | Zbl
,W.J. Hehre, L. Radom, P.V.R. Schleyer and J.A. Pople, Ab initio molecular orbital theory. Wiley (1986).
On the coupling of boundary integral and finite element methods. Math. Comput. 35 (1980) 1063–1079. | DOI | MR | Zbl
and ,Localized linear polynomial operators and quadrature formulas on the sphere. SIAM J. Numer. Anal. 47 (2008/09) 440–466. | MR | Zbl
and ,Finite difference model for infinite media. J. Eng. Mech. EMR 95 (1969) 859–877.
and ,Reappraisal of Laguerre type spectral methods. La Recherche Aerospatiale 6 (1985) 13–35. | MR | Zbl
, and ,Optimal numerical integration on a sphere. Math. Comput. 17 (1963) 361–383. | DOI | MR | Zbl
,C. Müller, Spherical harmonics. Vol. 17 of Lect. Notes Math. Springer-Verlag, Berlin (1966). | MR | Zbl
C. Müller, Analysis of Spherical Symmetries in Euclidean Spaces. Vol. 129 of Applied Mathematical Sciences. Springer (1998). | MR | Zbl
A. Ralston and Ph. Rabinowitz, A first course in numerical analysis, 2nd edition. Dover Publications, Inc., Mineola, New York (2001). | MR | Zbl
Spherical harmonics. Amer. Math. Monthly 73 (1966) 115–121. | DOI | MR | Zbl
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