Parameter-robust Uzawa-type iterative methods for double saddle point problems arising in Biot's consolidation and multiple-network poroelasticity models (original) (raw)
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Numerical Linear Algebra With Applications, 2019
The parameters in the governing system of partial differential equations of multicompartmental poroelastic models typically vary over several orders of magnitude making its stable discretization and efficient solution a challenging task. In this paper, inspired by the approach recently presented by Hong and Kraus [Parameter-robust stability of classical three-field formulation of Biot's consolidation model, ETNA (to appear)] for the Biot model, we prove the uniform stability, and design stable disretizations and parameter-robust preconditioners for flux-based formulations of multiple-network poroelastic systems. Novel parameter-matrix-dependent norms that provide the key for establishing uniform inf-sup stability of the continuous problem are introduced. As a result, the stability estimates presented here are uniform not only with respect to the Lamé parameter λ, but also with respect to all the other model parameters such as permeability coefficients Ki, storage coefficients cp i , network transfer coefficients βij , i, j = 1, • • • , n, the scale of the networks n and the time step size τ. Moreover, strongly mass conservative discretizations that meet the required conditions for parameterrobust stability are suggested and corresponding optimal error estimates proved. The transfer of the canonical (norm-equivalent) operator preconditioners from the continuous to the discrete level lays the foundation for optimal and fully robust iterative solution methods. The theoretical results are confirmed in numerical experiments that are motivated by practical applications.
Weakly Imposed Symmetry and Robust Preconditioners for Biot’s Consolidation Model
Computational Methods in Applied Mathematics
We discuss the construction of robust preconditioners for finite element approximations of Biot’s consolidation model in poroelasticity. More precisely, we study finite element methods based on generalizations of the Hellinger–Reissner principle of linear elasticity, where the stress tensor is one of the unknowns. The Biot model has a number of applications in science, medicine, and engineering. A challenge in many of these applications is that the model parameters range over several orders of magnitude. Therefore, discretization procedures which are well behaved with respect to such variations are needed. The focus of the present paper will be on the construction of preconditioners, such that the preconditioned discrete systems are well-conditioned with respect to variations of the model parameters as well as refinements of the discretization. As a byproduct, we also obtain preconditioners for linear elasticity that are robust in the incompressible limit.
Mathematical Methods in the Applied Sciences, 2016
In this paper, we consider a coupled system of mixed hyperbolic-parabolic type which describes the Biot consolidation model in poro-elasticity. We establish a local Carleman estimate for Biot consilidation system. Using this estimate, we prove the uniqueness and a Hölder stability in determining on the one hand a physical parameter arising in connection with secondary consolidation effects λ * and on the other hand the two spatially varying density by a single measurement of solution over ω × (0, T), where T > 0 is a sufficiently large time and a suitable subbdomain ω satisfying ∂ω ⊃ ∂Ω.
Numerical stabilization of Biot's consolidation model by a perturbation on the flow equation
International Journal for Numerical Methods in Engineering, 2008
In this paper a stabilized finite element scheme for the poroelasticity equations is proposed. This method, based on the perturbation of the flow equation, let us to use continuous piecewise linear approximation spaces for both displacements and pressure, obtaining solutions without oscillations independently of the chosen discretization parameters. The perturbation term depends on a parameter which is fixed in function of the size mesh and the properties of the material. In the one-dimensional case, this parameter is shown to be optimal. Some numerical experiments are presented indicating the efficiency of the proposed stabilization.
On stability and convergence of finite element approximations of Biot's consolidation problem
International Journal for Numerical Methods in Engineering, 1994
Stability and convergence analysis of finite element approximations of Biot's equations governing quasistatic consolidation of saturated porous media are discussed. A family of decay functions, parametrized by the number of time steps, is derived for the fully discrete backward Euler-Galerkin formulation, showing that the pore-pressure oscillations, arising from an unstable approximation of the incompressibility constraint on the initial condition, decay in time. Error estimates holding over the unbounded time domain for both semidiscrete and fully discrete formulations are presented, and a post-processing technique is employed to improve the pore-pressure accuracy.
A parallel-in-time fixed-stress splitting method for Biot's consolidation model
arXiv: Numerical Analysis, 2018
In this work, we study the parallel-in-time iterative solution of coupled flow and geomechanics in porous media, modelled by a two-field formulation of the Biot's equations. In particular, we propose a new version of the fixed stress splitting method, which has been widely used as solution method of these problems. This new approach forgets about the sequential nature of the temporal variable and considers the time direction as a further direction for parallelization. We present a rigorous convergence analysis of the method and a numerical experiment to demonstrate the robust behaviour of the algorithm.