Abstract
This paper is concerned with the implementation of efficient solution algorithms for elliptic problems with constraints. We establish theory which shows that including a simple scaling within well-established block diagonal preconditioners for Stokes problems can result in significantly faster convergence when applying the preconditioned MINRES method. The codes used in the numerical studies are available online.
Highlights
IntroductionThe motivation for this work is the development of fast and robust linear solvers for stabilized mixed approximations of the Stokes equations,
The motivation for this work is the development of fast and robust linear solvers for stabilized mixed approximations of the Stokes equations,−∇2v + ∇ p = f, −∇ · v = 0, together with suitable (Dirichlet, Neumann or mixed) boundary conditions
We suppose that the boundary value problem is discretized using standard mixed finite elements
Summary
The motivation for this work is the development of fast and robust linear solvers for stabilized mixed approximations of the Stokes equations,. There has been a great deal of research devoted to solving systems of the form (1.1) using preconditioned iterative methods; see [2] for a definitive review This body of work is relevant to any linear system that is generated by a mixed approximation; see [4, Chapter 3] for a characterization. Where H ∈ Rn×n is some symmetric positive definite approximation to the Schur complement S. In the specific case of the Stokes equations, the approximate Schur complement is either the mass matrix associated with the pressure approximation space. May and Moresi [16] scaled H = Q by the (fixed) viscosity of the fluid; the same scaling is applied in the Cahouet and Chabard preconditioner for generalized Stokes problems [5] None of these investigate the optimal choice of scaling parameter. The n × n matrix formed by extracting the diagonal of F ∈ Rn×n will be denoted diag(F)
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