Computation of the spectrum for some elliptic operators in periodic waveguides

In this paper we shall study a numerical method for computing the spectrum of some elliptic operators. It will be shown that the essential spectrum of an elliptic operator in an infinite periodic waveguide may contain gaps. Moreover, we construct examples where the number of band-gaps can be made arbitrarily large. Finally, we present some numerical examples by using the finite element method. The theoretical tool to accomplish this task is provided by the Krein-Birman-Vishik theory. Using the Gelfand transform we formulate the spectral problem for a quadratic pencil of formally self-adjoint operators on the periodicity cell. For fixed Gelfand parameter the model problem in the periodicity cell admits an increasing sequence of real and non-negative eigenvalues. We approximate these eigenvalues using the mixed finite element method using the Raviart-Thomas elements for the approximation of the eigenfunctions.