Convergence of the galerkin method for solving a nonlinear problem of the eigenmodes of microdisk lasers

Abstract

This paper investigates an eigenvalue problem for the Helmholtz equation on the plane modeling the laser radiation of two-dimensional microdisk resonators. It was reduced to an eigenvalue problem for a holomorphic Fredholm operator-valued function A(k). For its numerical solution, the Galerkin method was proposed, and its convergence was proved. Namely, a sequence of the finite-dimensional holomorphic operator functions An(k) that converges regularly to A(k) was constructed. Further, it was established that there is a sequence of eigenvalues kn of the operator-valued functions An(k) converging to k0 for each eigenvalue k0 of the operator-valued function A(k). If {kn}n∈N is a sequence of eigenvalues of the operator-valued functions An(k) converging to a number of k0, then k0 is an eigenvalue of A(k). The estimates for the rate of convergence of {kn}n∈N to k0 depend either on the order of the pole k0 of the operator-valued function A−1(k), or on the algebraic multiplicities of all eigenvalues of An(k) in a neighborhood of k0, or on the number of different eigenvalues of An(k) in this neighborhood. The reasoning is based on the fundamental results of the theory of holomorphic operator-valued functions and is important for the theory of mic-rodisk lasers, because it significantly expands the class of devices interesting for applications that allow mathematical modeling based on numerical methods that are strictly justified.