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 a s the thickness tends to zero. Consequently, the membrane approximatio n gives a good description of vibration properties of a thin shell. Th is vibration problem is associated with a non-compact resolvent operat or, and spectral pollution could appear when computing Galerkin approx imations. That is to say, there could exist sequences of eigenvalues o f the approximate problems that converge to points of the resolvent se t 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 l ocalization 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.