A general adaptive finite element eigen-algorithm stemming from Wittrick-Williams algorithm

Abstract Stemming from Wittrick-Williams algorithm for transcendental eigenproblems, a general adaptive finite element (FE) procedure is proposed for general linear eigenproblems. The pivot of the proposed procedure is the counting method based on Sturm sequences, a simplified version of the Wittrick-Williams algorithm for linear matrix eigenproblems, based on which a general eigen-algorithm is developed by integrating a number of effective approaches such as a two-phased divide-and-conquer scheme, a guided-and-guarded technique for eigenvalue seeking, the inverse and subspace iterations for mode finding, etc. The adopted error estimator in the adaptive FE procedure is the recently developed one which is based on the element energy projection (EEP) technique for pointwise error estimation and hence is able to control errors by maximum norm with local mesh refinement. Representative numerical examples of plates and shells are given to show the validity, reliability and efficiency of the proposed procedure.