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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.

MSC:
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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