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Convergence analysis of the Newton-Schur method for the symmetric elliptic eigenvalue problem

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 $Ω$ into the nonlinear eigenvalue subproblem on $Γ$, which is the union of subdomain boundaries. We prove that the convergence rate for the Newton-Schur method is $ε_{N}\leq CH^{2}(1+\ln(H/h))^{2}ε^{2}$, where the constant $C$ is independent of the fine mesh size $h$ and coarse mesh size $H$, and $ε_{N}$ and $ε$ are errors after and before one iteration step respectively. Numerical experiments confirm our theoretical analysis.