Error estimation of eigenfrequencies for elasticity and shell problems

In this paper, a method for deriving computable estimates of the approximation error in eigenvalues or eigenfrequencies of three-dimensional linear elasticity or shell problems is presented. The analysis for the error estimator follows the general approach of goal-oriented error estimation for which the error is estimated in so-called quantities of interest, here the eigenfrequencies, rather than global norms. A general theory is developed and is then applied to the linear elasticity equations. For the shell analysis, it is assumed that the shell model is not completely known and additional errors are introduced due to modeling approximations. The approach is then based on recovering three-dimensional approximations from the shell eigensolution and employing the error estimator developed for linear elasticity. The performance of the error estimator is demonstrated on several test problems.