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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
BiCGstab
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