An augmented Lagrangian approach to the numerical solution of a non-smooth eigenvalue problem

In this article, we address the numerical solution of a non- smooth eigenvalue problem, which has implications in plasticity theory and image processing. The smallest eigenvalue of the nonsmooth operator under consideration is shown to be the same for all bounded, sufficiently smooth, domains in two space dimensions. Piecewise linear finite elements are used for the discretization of eigenfunctions and eigenvalues. An augmented Lagrangian method is proposed for the computation of the minima of the associated non-convex optimization problem. The convergence of finite element approximations of generalized eigenpairs is investigated. Numerical solutions are presented for the first eigenvalue and eigenfunction. For non-simply connected domains, the augmented Lagrangian method also captures larger eigenvalues as local minima. Bifurcation between the first and second eigenvalues is investigated numerically.