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 system matrix is symmetric, but indefinite, so variational convergence theory usually used to prove multigrid cannot applied. In 2016 paper Numerische Mathematik Notay has presented different algebraic approach analyzes properly preconditioned saddle problems, proving two-grid method. The present where blocks are circulant within this framework. It contains sufficient conditions for and optimal parameters preconditioning unilevel multilevel problem smoother used. analysis based on generating symbols blocks. Further, it shown structure kept coarse level, allowing recursive application W- or V-cycle studying “level independency” property. Numerical results demonstrate efficiency proposed method Toeplitz case.