Symbol based convergence analysis in multigrid methods for saddle point problems

TL;DR

This paper extends convergence analysis of multigrid methods for saddle point problems with circulant blocks, providing conditions for optimal preconditioning and demonstrating recursive application with numerical validation.

Contribution

It introduces a symbol-based analysis framework for circulant saddle point problems, deriving convergence conditions and optimal parameters for multigrid methods.

Findings

Derived sufficient conditions for convergence.

Identified optimal preconditioning parameters.

Validated efficiency through numerical experiments.

Abstract

Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the Two-Grid method. In the present paper we analyze saddle point problems where the blocks are circulant within this framework. We are able to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, we show that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W- or V-cycle and proving the "level independency" property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.