On Schwarz-type Smoothers for Saddle Point Problems

Abstract

In this paper we consider additive Schwarz-type iteration methods for saddle point problems as smoothers in a multigrid method. Each iteration step of the additive Schwarz method requires the solutions of several small local saddle point problems. This method can be viewed as an additive version of a (multiplicative) Vanka-type iteration, well-known as a smoother for multigrid methods in computational fluid dynamics. It is shown that, under suitable conditions, the iteration can be interpreted as a symmetric inexact Uzawa method. In the case of symmetric saddle point problems the smoothing property, an important part in a multigrid convergence proof, is analyzed for symmetric inexact Uzawa methods including the special case of the additive Schwarz-type iterations. As an example the theory is applied to the Crouzeix-Raviart mixed finite element for the Stokes equations and some numerical experiments are presented.

Multigrid analysis for the time dependent Stokes problem

“…The analysis of various multigrid methods for the Stokes problem (α = 0) can be found in several papers; see [5,8,21,26,29,33]. The smoothing analysis from [6,27,37] can also be merged with the approximation property from [19,33] to establish the convergence of the two-grid method for the case of α = 0.…”

Finite Element Approximation of a Three Dimensional Phase Field Model for Void Electromigration

“…[31], where convergence for a multigrid method employing a Jacobi smoother is shown. Moreover, the convergence of a large class of block Gauss-Seidel smoothers for saddle-point problems arising from the discretizations of Stokes and NavierStokes equations has recently been shown in [25].…”

Optimal multigrid preconditioned semi‐monotonic augmented Lagrangians applied to the Stokes problem

“…Moreover, if Kernel(B T ) = {0}, then there is also a unique Lagrange multiplier p * ∈ Q. This type of problems arises in mixed variational formulations of the Stokes problem [3], elliptic partial differential equations (pde) with periodic boundary conditions [4], applications of the domain decomposition methods to parallel solution of three-dimensional elasticity problems [5], modeling of laminated composites [6], development of scalable algorithms for parallel solution of variational inequalities [7], or large-scale optimization [8]. The results involving the solution of (1) are also useful in solution of problems with bound and inequality constraints [9].…”

“…Recently, in [15,16] the algorithm has been modified and proven to converge efficiently even with a relatively poor preconditioning of S. Another approach, which is based on multigrid methods as an outer iteration combined with appropriate smoothers as a sort of inner iteration, was proposed in [17]. In [3], a multigrid method with an additive Schwarz smoother is proposed and analyzed with an application to the Stokes problem. It is proven that the method is equivalent to the symmetric Uzawa algorithm.…”