Convergence Analysis of Newton–Schur Method for Symmetric Elliptic Eigenvalue Problem

Abstract

In this paper, we consider the Newton–Schur method in Hilbert space and obtain quadratic convergence. For the symmetric elliptic eigenvalue problem discretized by the standard finite element method and nonoverlapping domain decomposition method, we use the Steklov–Poincaré operator to reduce the eigenvalue problem on the domain into the nonlinear eigenvalue subproblem on , which is the union of subdomain boundaries. We prove that the convergence rate for the Newton–Schur method is , where the constant is independent of the fine mesh size and coarse mesh size , and and are errors after and before one iteration step, respectively. For one specific inner product on , a sharper convergence rate is obtained, and we can prove that . Numerical experiments confirm our theoretical analysis.