On a variational approximation method for a class of elliptic eigenvalue problems in composite structures

Abstract

We consider a second-order elliptic eigenvalue problem on a convex polygonal domain, divided in M M nonoverlapping subdomains. The conormal derivative of the unknown function is continuous on the interfaces, while the function itself is discontinuous. We present a general finite element method to obtain a numerical solution of the eigenvalue problem, starting from a nonstandard formally equivalent variational formulation in an abstract setting in product Hilbert spaces. We use standard Lagrange finite element spaces on the subdomains. Moreover, the bilinear forms are approximated by suitable numerical quadrature formulas. We obtain error estimates for both the eigenfunctions and the eigenvalues, allowing for the case of multiple exact eigenvalues, by a pure variational method.