Finite element approximation to the extremal eigenvalue problem for inhomogenous materials

Abstract

In this paper, an extremal eigenvalue problem corresponding to an inhomogeneous membrane which is composed of two different materials with different densities is investigated. The convergence of the finite element discretization and the error order for the smallest eigenvalue are obtained. A monotonic decreasing algorithm is presented to solve the discretized problem and numerical examples are given to demonstrate the error estimation as well as the efficiency of the method.

Extremal Eigenvalues of the Sturm-Liouville Problems with Discontinuous Coefficients

“…Proof. Firstly we claim that Ph P* weak star in L°°(O,Ly), inThe proof of this claim is similar to Theorem 3.8 in the paper[24] and Lemma 2.3. Firstly passage to a subsequence we have p* -* p* weak star in L°°(O,L^), A*^(p*) -^ X* and Vft(pp ^ V* weakly in H\O, Ly).…”

“…The computational result can also be found in Nemat-Nasser et al [27,28]. Recently, Liang et al [24] study the convergence of the finite element method for the extremal eigenvalue problem with variable density function. Inspired by the previous works, we exploit Extremal Eigenvalues of the Sttirm-Iiouville Problems with Discontinuous Coefficients 659 the finite element method in the extremal eigenvalue problems with variable conductivities in the one dimension interval.…”

“…[18]), Therefore, Problem 1.1 can be equivalently presented as min Similar as in [24], we may propose a monotonie decreasing algorithm for Problem 1.1, see Algorithm 2.1. It can be shown that the above algorithm produces a monotonie decreasing eigenvalue sequence, and it converges to a fixed point of the composite map a ^ GOU.…”