A Full Multigrid Method for Nonlinear Eigenvalue Problems

TL;DR

This paper introduces a full multigrid method for nonlinear eigenvalue problems that transforms the problem into linear boundary value problems and nonlinear eigenvalue problems across multiple finite element spaces, achieving optimal computational efficiency.

Contribution

The paper presents a novel multigrid scheme that efficiently solves nonlinear eigenvalue problems by reducing them to linear boundary value problems and coarser nonlinear eigenvalue problems, with proven optimal computational work.

Findings

The new method is computationally optimal, matching the cost of solving linear boundary value problems.

Numerical experiments confirm the efficiency and effectiveness of the proposed multigrid approach.

The scheme improves the overall efficiency of solving nonlinear eigenvalue problems.

Abstract

This paper is to introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We will prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.