## A preconditioned iterative method for saddlepoint problems.(English)Zbl 0760.65033

The authors discuss the convergence and the preconditioning of the minimal residual method applied to the symmetric, but indefinite system $A{x\choose y}={b\choose c},\quad \text{with}\quad A=\bigl({M\atop B^ T}{B\atop 0}\bigr),$ where $$M\in\mathbb{R}^{n\times n}$$ is symmetric and positive definite, $$B\in\mathbb{R}^{n\times m}$$ with $$m\leq n$$, and $$\text{rank}(B)=m$$. The convergence analysis is based on a precise estimate of the negative and positive parts of the spectrum of $$A$$. The preconditioned system has the form $$S^{-1}AS$$, where the preconditioner $$S=\text{diag}(L,R^ T)$$ is defined by appropriately chosen preconditioning blocks $$L$$ and $$R$$ of the dimension $$n\times n$$ and $$m\times m$$, respectively.
The method is applied to the solution of mixed finite element schemes arising from the Stokes equation and from second order elliptic boundary value problems. Numerical results are presented.

### MSC:

 65F10 Iterative numerical methods for linear systems 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65F35 Numerical computation of matrix norms, conditioning, scaling 35J25 Boundary value problems for second-order elliptic equations 35Q30 Navier-Stokes equations
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