A Numerical Solution for the Nonlinear Eigenvalue System

Abstract

In the application of the classical techniques of the calculus of variations to the nonlinear optimization problem, the nonlinear eigenvalue problem -λ • AU = f (U), where △ is an elliptic differential operator, is often encountered. A new numerical technique, using a finite-element-based algorithm, for finding the solution U(X) where X ∈ R ⁿ the eigenvalue λ, and the maximum of the corresponding cost functional J(U) where J(U)= f _Ω F(U)dx $f(U) = \frac{{\partial F(U)}}{{\partial U}}$ is presented. This algorithm is applicable to a class of problems for which the nonlinear function f (U) has specific properties. The algorithm is simple but fast and both convergence and uniqueness of the solution are demonstrated in several examples.