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

TL;DR

This paper analyzes the quadratic convergence of the Newton-Schur method applied to symmetric elliptic eigenvalue problems, using domain decomposition and finite element discretization, with theoretical proofs and numerical validation.

Contribution

It provides a convergence analysis of the Newton-Schur method in Hilbert space for discretized elliptic eigenvalue problems, including explicit error bounds and independence from mesh sizes.

Findings

Quadratic convergence rate established theoretically.

Error bound depends on coarse mesh size H and fine mesh size h.

Numerical experiments confirm theoretical predictions.

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 $\epsilon_{N}\leq CH^{2}(1+\ln(H/h))^{2}\epsilon^{2}$, 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.