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

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 $A_n(k)$ that converges regularly to $A(k)$ was constructed. Further, it was established that there is a sequence of eigenvalues $k_n$ of the operator-valued functions $A_n(k)$ converging to $k_0$ for each eigenvalue $k_0$ of the operator-valued function $A(k)$. If $\{k_n \}_{n \in {N}}$ is a sequence of eigenvalues of the operator-valued functions $A_n(k)$ converging to a number of $k_0$, then $k_0$ is an eigenvalue of $A(k)$. The estimates for the rate of convergence of $\{k_n\} _ {n \in {N}}$ to $k_0$ depend either on the order of the pole $k_0$ of the operator-valued function $A^{-1}(k)$, or on the algebraic multiplicities of all eigenvalues of $A_n(k)$ in a neighborhood of $k_0$, or on the number of different eigenvalues of $A_n(k)$ 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 microdisk lasers, because it significantly expands the class of devices interesting for applications that allow mathematical modeling based on numerical methods that are strictly justified.