Convergence analysis of the Newton-Schur method for the 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 non-overlapping domain decomposition method, we use the Steklov-Poincaré operator to reduce the eigenvalue problem on the domain $\Omega$ into the nonlinear eigenvalue subproblem on $\Gamma$, which is the union of subdomain boundaries. We prove that the convergence rate for the Newton-Schur method is ${\epsilon }_N\le C{H}^2(1+\ln (H/h){)}^2{\epsilon }^2$, where the constant $C$ is independent of the fine mesh size $h$ and coarse mesh size $H$, and ${\epsilon }_N$ and $\epsilon$ are errors after and before one iteration step respectively. Numerical experiments confirm our theoretical analysis.