Numerical Approximation of Second-Order Elliptic Problems in Unbounded Domains

This paper deals with the numerical resolution of elliptic problems in unbounded domains using inverted finite elements. In opposition to conventional approaches which are based on the truncation of the domain, the suggested method keeps the domain unbounded and is based on a description of the asymptotic behavior in an appropriate functional framework. The method and its mathematical properties are presented first, and some computational examples are carried out. The obtained numerical results demonstrate the efficiency of the method.

[1]  A. Toselli.  A new family of exponential infinite elements for the analysis of lossless electromagnetic waveguides , 1997 .

[2]  B. Hanouzet.  Espaces de Sobolev avec poids. Application au problème de Dirichlet dans un demi espace , 1971 .

[3]  M. M. Babatin,et al.  Inverted finite elements for degenerate and radial elliptic problems in unbounded domains , 2015 .

[4]  S. Tsynkov.  Numerical solution of problems on unbounded domains. a review , 1998 .

[5]  Vivette Girault,et al.  Weighted Sobolev spaces for Laplace's equation in Rn , 1994 .

[6]  V. A. Kondrat'ev,et al.  Boundary problems for elliptic equations in domains with conical or angular points , 1967 .

[7]  J. Giroire.  Etude de quelques problèmes aux limites extérieurs et résolution par équations intégrales , 1987 .

[8]  A. Bayliss,et al.  Radiation boundary conditions for wave-like equations , 1980 .

[9]  T. Boulmezaoud.  On the Stokes system and on the biharmonic equation in the half‐space: an approach via weighted Sobolev spaces , 2002 .

[10]  T. Boulmezaoud,et al.  Weighted Lp theory of the Stokes and the bilaplacian operators in the half-space , 2008 .

[11]  T. Boulmezaoud.  Inverted finite elements: A new method for solving elliptic problems in unbounded domains , 2005 .

[12]  Problème de Stokes dans ℝn et espaces de Sobolev avec poids , 1997 .

[13]  C. Brebbia,et al.  Boundary Element Techniques , 1984 .

[14]  Gernot Beer,et al.  The Boundary Element Method with Programming: For Engineers and Scientists , 2008 .

[15]  David S. Burnett,et al.  A three‐dimensional acoustic infinite element based on a prolate spheroidal multipole expansion , 1994 .

[16]  Leszek Demkowicz,et al.  Solution of 3D-Laplace and Helmholtz equations in exterior domains using hp-infinite elements , 1996 .

[17]  T. Boulmezaoud,et al.  Eigenfunctions of a weighted Laplace operator in the whole space , 2013 .

[18]  K. Gerdes,et al.  A REVIEW OF INFINITE ELEMENT METHODS FOR EXTERIOR HELMHOLTZ PROBLEMS , 2000 .

[19]  T. Boulmezaoud.  On the Laplace operator and on the vector potential problems in the half‐space: an approach using weighted spaces , 2003 .

[20]  O. C. Zienkiewicz,et al.  Diffraction and refraction of surface waves using finite and infinite elements , 1977 .

[21]  F. Ihlenburg.  Finite Element Analysis of Acoustic Scattering , 1998 .

[22]  U. Razafison,et al.  On the steady Oseen problem in the whole space , 2005 .

[23]  A. Majda,et al.  Radiation boundary conditions for acoustic and elastic wave calculations , 1979 .