On the numerical approximation of some types of nonstandard second-order eigenvalue problems for vector valued functions

In this paper we consider some types of second-order elliptic eigenvalue problems (EVPs) for vector valued functions on a convex polygonal domain in the plane, with nonstandard boundary conditions (BCs) of nonlocal type. The aim of the paper is twofold. First, we pass to a variational form of the EVP, which is shown to be formally equivalent to the differential EVP and which is proved to fit into the well-known general framework of abstract elliptic EVPs for bilinear forms in Hilbert spaces, treated, e.g., in Raviart, Thomas, Introduction à lʹanalyse numérique des équations aux dérivées partielles, 3rd Edition, Masson, Paris, 1992. This implies the existence of exact eigenpairs with suitable properties. Next, we study finite element approximation methods for this problem. We argue that similar convergence results and error estimates hold as those established, e.g., in Dautray, Lions, Analyse numérique et calcul numérique pour les sciences et les techniques, tome 2, Masson, Paris, 1985, Chapitre 12, Raviart, Thomas, Introduction à lʹanalyse numérique des équations aux dérivées partielles, 3rd Edition, Masson, Paris, 1992 or Strang, Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973 for elliptic EVPs for a scalar function, with classical local BCs of Dirichlet, Neumann or Robin type. Here the nonlocal character of the BCs constitutes a major difficulty in the analysis, requiring the introduction and error estimation of a new, suitably modified (vector) Lagrange interpolant on the FE-mesh. The theoretical error estimate for the eigenvalues is confirmed by an illustrative numerical example.