Convergence analysis of the Newton-Schur method for the symmetric elliptic eigenvalue problem
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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≤ CH^2(1+ln(H/h))^2ϵ^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.
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