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An optimal preconditioner for a class of saddle point problems with a penalty term. (English) Zbl 0912.65018
Iterative methods are considered for a class of symmetric indefinite linear systems \(Ax=b\) arising from finite element discretizations of certain elliptic problems. The conjugate gradient method dealing with the preconditioned system \(B^{-1}Ax=B^{-1}b\) is described. The author analyzes a known convergence estimate for the method that involves the condition number \(\kappa(B^{-1}A)={{\max \{ | \lambda | : \lambda \in \sigma(B^{-1}A)\}} \over {\min \{ | \lambda | : \lambda \in \sigma(B^{-1}A) \}}}\) of the preconditioned system, where \(\sigma(B^{-1}A)\) denotes the spectrum of \(B^{-1}A\). He obtains the upper estimate for \(\kappa(B^{-1}A)\) in terms of characteristics of the original differential problem. In conclusion, numerical results for problems of planar linear elasticity are discussed.

65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
74B05 Classical linear elasticity
74S05 Finite element methods applied to problems in solid mechanics
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