A full multigrid method for the Steklov eigenvalue problem

ABSTRACT

This paper introduces a kind of parallel multigrid method for solving Steklov eigenvalue problem based on the multilevel correction method. Instead of the common costly way of directly solving the Steklov eigenvalue problem on some fine space, the new method contains some boundary value problems on a series of multilevel finite element spaces and some steps of solving Steklov eigenvalue problems on a very low dimensional space. The linear boundary value problems are solved by some multigrid iteration steps. We will prove that the computational work of this new scheme is truly optimal, the same as solving the corresponding linear boundary value problem. Besides, this multigrid scheme has a good scalability by using parallel computing technique. Some numerical experiments are presented to validate our theoretical analysis.

KEYWORDS:

• Steklov eigenvalue problem
• multigrid method
• multilevel correction
• parallel computing
• finite element method

2010 AMS SUBJECT CLASSIFICATIONS:

• 65F15
• 65N15
• 65N25
• 65N30
• 65N50

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by the National Natural Science Foundations of China (NSFC) [11801021]. The Foundation for Connotation Development of Beijing University of Technology [107000514117002], Foundation for Fundamental Research of Beijing University of Technology [006000546318504].