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Robust preconditioning for stochastic Galerkin formulations of parameter-dependent nearly incompressible elasticity equations. (English) Zbl 1428.65087

The paper deals with the numerical solution of nearly incompressible elasticity problems as the Herrmann formulation of linear elasticity. The Young’s modulus \(E\) is modeled as spatially varying random field. The new three-field mixed formulation and its well-posedness are presented. The problem is discretized by the stochastic Galerkin finite element method, the existence of the approximate solution is proved. Further, the paper focuses on the efficient solution of the arising indefinite linear algebraic system. A new preconditioner is introduced for the use of the minimal residual method. Eigenvalue bounds for the preconditioned system are established and shown to be independent of the discretization parameters and the Poisson ratio. Numerical experiments, supporting the theoretical results, are presented.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
35R60 PDEs with randomness, stochastic partial differential equations
74B05 Classical linear elasticity
35Q74 PDEs in connection with mechanics of deformable solids
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References:

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