Symbol based convergence analysis in multigrid methods for saddle point problems

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. The present paper analyzes saddle point problems where the blocks are circulant within this framework. It contains sufficient conditions for convergence and optimal parameters for the preconditioning of the unilevel and multilevel saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, it is shown 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 studying the “level independency” property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.