# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## A locally optimal preconditioned Newton-Schur method for symmetric elliptic eigenvalue problemsHTML articles powered by AMS MathViewer

by Wenbin Chen, Nian Shao and Xuejun Xu
Math. Comp. 92 (2023), 2655-2684
DOI: https://doi.org/10.1090/mcom/3860
Published electronically: July 5, 2023

## Abstract:

A locally optimal preconditioned Newton-Schur method is proposed for solving symmetric elliptic eigenvalue problems. Firstly, the Steklov-Poincaré operator is used to project the eigenvalue problem on the domain $\Omega$ onto the nonlinear eigenvalue subproblem on $\Gamma$, which is the union of subdomain boundaries. Then, the direction of correction is obtained via applying a non-overlapping domain decomposition method on $\Gamma$. Four different strategies are proposed to build the hierarchical subspace $U_{k+1}$ over the boundaries, which are based on the combination of the coarse-subspace with the directions of correction. Finally, the approximation of eigenpair is updated by solving a local optimization problem on the subspace $U_{k+1}$. The convergence rate of the locally optimal preconditioned Newton-Schur method is proved to be $\gamma =1-c_{0}T_{h,H}^{-1}$, where $c_{0}$ is a constant independent of the fine mesh size $h$, the coarse mesh size $H$ and jumps of the coefficients; whereas $T_{h,H}$ is the constant depending on stability of the decomposition. Numerical results confirm our theoretical analysis.
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Bibliographic Information
• Wenbin Chen
• Affiliation: School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, People’s Republic of China
• MR Author ID: 653643
• ORCID: 0000-0001-8305-0764
• Email: wbchen@fudan.edu.cn
• Nian Shao
• Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
• MR Author ID: 1410652
• Email: nshao20@fudan.edu.cn
• Xuejun Xu
• Affiliation: School of Mathematical Sciences, Tongji University, Shanghai 200442, People’s Republic of China; and LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
• MR Author ID: 365400
• Email: xxj@lsec.cc.ac.cn
• Received by editor(s): May 10, 2022
• Received by editor(s) in revised form: May 10, 2022, and February 14, 2023
• Published electronically: July 5, 2023
• Additional Notes: The first author was supported by the National Natural Science Foundation of China (NSFC) 12241101 and the National Key R&D Program of China (2019YFA0709502). The third author was supported by the National Natural Science Foundation of China (grants 12071350), Shanghai Municipal Science and Technology Major Project 2021SHZDZX0100, and Science and Technology Commission of Shanghai Municipality.
The first author is the corresponding author.

• Dedicated: This paper is dedicated to the memory of Prof. Zhongci Shi