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

DOI10.1137/21M1448847zbMath1519.65051arXiv2205.01861MaRDI QIDQ6398208

Nian Shao, Wenbin Chen

Publication date: 3 May 2022

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-Poincar'e 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 $epsilo{n}_{N}leqC{H}^2(1+ln(H/h){)}^{2}epsilo{n}^2$, where the constant $C$ is independent of the fine mesh size $h$ and coarse mesh size $H$, and $epsilo{n}_N$ and $epsilon$ are errors after and before one iteration step respectively. Numerical experiments confirm our theoretical analysis.

Mathematics Subject Classification ID

Multigrid methods; domain decomposition for boundary value problems involving PDEs (65N55) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Numerical methods for eigenvalue problems for boundary value problems involving PDEs (65N25)

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