In this paper we present a finite element method (FEM) to a nonstandard second-order elliptic eigenvalue problem defined on a two-component domain consisting of two intervals with a contact point. This vector problem involves a nonlocal (integral type) coupling condition between the solution components. By introducing suitable degrees of freedom for the quadratic finite element and a corresponding vector Lagrange interpolant we derive optimal order finite element approximation.
Some numerical aspects concerning the method implementation are considered. Illustrative example is given which shows the efficiency of the proposed method.