Eigenvalues of the Laplacian through boundary integral equations

A numerical method for the eigenvalue problem of the Laplacian in two-dimensional domains is developed in this paper. This method requires $O( N )$ operations for calculating one eigenvalue in each iteration step, where N is the number of boundary points in the discretization. It is based on the boundary integral formulation which reduces the computation of the eigenvalues to the zeros of the function $\mu _1 ( \lambda )$ defined as the smallest eigenvalue of a related matrix. Iteration methods such as the Lanczos method are used to compute $\mu_1 ( \lambda )$, which requires the multiplication of an $N \times N$ matrix with a vector. The multipole expansion techniques developed for the potential problems by Rokhlin [J. Comput. Phys., 60 (1985), pp. 187–207] are applied and extended here, and the number of operations is reduced to $O( N )$ for this multiplication. The zeros of $\mu _1 ( \lambda )$ are found by the method of quadratic interpolation. A method for finding the kth eigenvalue with the value of k prespecified is also presented. It is based on continuously tracing the eigenvalue while the domain is deforming to (or from) the unit disk. Only five values of $\mu _1 $ are required in this tracing process.