TWO-SIDED BOUNDS OF EIGENVALUES USING CONFORMING AND NONCONFORMING FINITE ELEMENTS

Abstract

The main goal of this paper is to present an original algorithm andnumerical approach which gives two-sided bounds of eigenvalues for second-order ellipticoperator. The method consists of finite element solving of the problem, making a choiceof conforming elements and then constructing corresponding nonconforming interpolantof the approximate conforming eigenfunctions. Thus, solving only once the eigenvalueproblem, we get upper and lower bounds for the exact eigenvalues. For this purpose weapply integral type finite elements, which use integral values on their edges or/and onthe elements itselfs as degrees of freedom. From a practical point of view our aim is touse lowest possible order finite elements. Furthermore, the fact that the nonconforminginterpolants use the nodal values of the conforming approximate eigenfunctions givesan obvious computational advantage.

Computational aspects of the algorithm are discussed. Finally, numerical experi-ments are also provided.