A Multilevel Newton Iteration Method for Eigenvalue Problems

TL;DR

This paper introduces a multilevel Newton iteration method that enhances the efficiency of solving eigenvalue problems by combining coarse space eigenproblems with Newton-based linear solves, validated through numerical examples.

Contribution

It presents a novel multilevel Newton iteration approach that reduces computational cost for eigenvalue problems by integrating coarse eigenproblems with Newton steps.

Findings

Improved computational efficiency demonstrated in numerical tests.

Effective reduction in problem size for eigenvalue computations.

Validation of the method's accuracy and speed.

Abstract

We propose a new type of multilevel method for solving eigenvalue problems based on Newton iteration. With the proposed iteration method, solving eigenvalue problem on the finest finite element space is replaced by solving a small scale eigenvalue problem in a coarse space and solving a series of augmented linear problems, derived by Newton step in the corresponding series of finite element spaces. This iteration scheme improves overall efficiency of the finite element method for solving eigenvalue problems. Finally, some numerical examples are provided to validate the efficiency of the proposed numerical scheme.