Extremal Eigenvalues of the Sturm-Liouville Problems with Discontinuous Coefficients

Abstract

In this paper, an extremal eigenvalue problem to the Sturm-Liouville equations with discontinuous coefficients and volume constraint is investigated. Liouville transformation is applied to change the problem into an equivalent minimization problem. Finite element method is proposed and the convergence for the finite element solution is established. A monotonie decreasing algorithm is presented to solve the extremal eigenvalue problem. A global convergence for the algorithm in the continuous case is proved. A few numerical results are given to depict the efficiency of the method. conductivities, the first eigenvalue and its corresponding eigenfunction represents the first mode of heat diffusion pattern. In particular, we assume that the heat conductor is made by two materials with conductivities a and ß, 0 < a < ß. Let the materials with the conductivity a occupies a measurable set D c f2, then where XD is the characteristic function of D. The material with conductivity a are assumed to have a fixed volume, which leads to the following optimization problem:where J4 is the admissible set for all possible choices of the conductivity function which is defined as: D, f cr = c j-.where J is the average of integral function on the domain and c is a constant which satisfies:a<c<ß.F{t*) = argmax] phV^dy -t* PhVhqWdy \ = 0, U J

Finite element approximation to the extremal eigenvalue problem for inhomogenous materials

“…But the finite element analysis on extremal eigenvalue problem is very limited. In [15], convergence of finite element method on extremal eigenvalue problem for one dimensional domain is provided. However to the best of our knowledge, there is no such analysis in a two or three dimensional domain.…”