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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.

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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