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

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.