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Parameter-robust Uzawa-type iterative methods for double saddle point problems arising in Biot’s consolidation and multiple-network poroelasticity models. (English) Zbl 1471.65143

The authors consider the stationary iterative methods for solving the equations of multiple-network poroelastic theory which describe flow in deformable porous media. They propose and analyze a class of fully decoupled iterative schemes. The proposed method fully decouples the fluid velocity, fluid pressure and solid displacement fields, contrary to the fixed-stress iterative scheme, which decouples only the flow from the mechanics problem. In every iteration, the smaller subsystems are solved. The authors present convergence analysis which proves the parameter-robust linear convergence of the new algorithm. The rate of contraction is strictly less than one independent of all physical and discretization parameters. The theoretical results are confirmed by a series of numerical tests.
Reviewer: Yan Xu (Hefei)

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F08 Preconditioners for iterative methods
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
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