Numerische Mathematik

, Volume 75, Issue 4, pp 473-500

First online:

On spectral pollution in the finite element approximation of thin elastic “membrane” shells

  • J. RappazAffiliated withDépartement de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
  • , J. Sanchez HubertAffiliated withUniversité de Caen, U.F.R. de Sciences, Mécanique, Esplanade de la Paix, F-14032 Caen, France
  • , E. Sanchez PalenciaAffiliated withLaboratoire de Modélisation en Mécanique, Université Pierre et Marie Curie et CNRS, 4, place Jussieu, F-75252 Paris, France; e-mail: LMM@CICRP.JUSSIEU.FR
  • , D. VassilievAffiliated withSchool of Mathematics, University of Sussex, Falmer, Brighton BN 1 9 QH, Great Britain

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The bending terms in a shell are small with respect to membrane ones as the thickness tends to zero. Consequently, the membrane approximation gives a good description of vibration properties of a thin shell. This vibration problem is associated with a non-compact resolvent operator, and spectral pollution could appear when computing Galerkin approximations. That is to say, there could exist sequences of eigenvalues of the approximate problems that converge to points of the resolvent set of the exact problem. We give an account of the state of the art of this problem in shell theory. A description of the phenomenon and its interpretation in terms of spectral families are given. A theorem of localization of the region where pollution may appear is stated and its complete proof is published for the first time. Recipes are given for avoiding the pollution as well as indications on the possibility of a posteriori elimination.

Mathematics Subject Classification (1991):65N30