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Register a new userParameter-robust convergence analysis of fixed-stress split iterative method for multiple-permeability poroelasticity systems
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We consider flux-based multiple-porosity/multiple-permeability poroelasticity systems describing multiple-network flow and deformation in a poro-elastic medium, sometimes also referred to as MPET models. The focus of the paper is on the convergence analysis of the fixed-stress split iteration, a commonly used coupling technique for the flow and mechanics equations in poromechanics. We formulate the fixed-stress split method in the present context and prove its linear convergence. The contraction rate of this fixed-point iteration does not depend on any of the physical parameters appearing in the model. The theory is confirmed by numerical results which further demonstrate the advantage of the fixed-stress split scheme over a fully implicit method relying on norm-equivalent preconditioning.
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This work is concerned with the iterative solution of systems of quasi-static multiple-network poroelasticity (MPET) equations describing flow in elastic porous media that is permeated by single or multiple fluid networks. Here, the focus is on a three-field formulation of the problem in which the displacement field of the elastic matrix and, additionally, one velocity field and one pressure field for each of the $n ge 1$ fluid networks are the unknown physical quantities. Generalizing Biots model of consolidation, which is obtained for $n=1$, the MPET equations for $nge1$ exhibit a double saddle point structure. The proposed approach is based on a framework of augmenting and splitting this three-by-three block system in such a way that the resulting block Gauss-Seidel preconditioner defines a fully decoupled iterative scheme for the flux-, pressure-, and displacement fields. In this manner, one obtains an augmented Lagrangian Uzawa-type method, the analysis of which is the main contribution of this work. The parameter-robust uniform linear convergence of this fixed-point iteration is proved by showing that its 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 that compare the new fully decoupled scheme to the very popular partially decoupled fixed-stress split iterative method, which decouples only flow--the flux and pressure fields remain coupled in this case--from the mechanics problem. We further test the performance of the block triangular preconditioner defining the new scheme when used to accelerate the GMRES algorithm.
We present an iterative coupling scheme for the numerical approximation of the mixed hyperbolic-parabolic system of fully dynamic poroelasticity. We prove its convergence in the Banach space setting for an abstract semi-discretization in time that allows the application of the family of diagonally implicit Runge-Kutta methods. Recasting the semi-discrete solution as the minimizer of a properly defined energy functional, the proof of convergence uses its alternating minimization. The scheme is closely related to the undrained split for the quasi-static Biot system.
Linear poroelasticity models have a number of important applications in biology and geophysics. In particular, Biots consolidation model is a well-known model that describes the coupled interaction between the linear response of a porous elastic medium and a diffusive fluid flow within it, assuming small deformations. Although deterministic linear poroelasticity models and finite element methods for solving them numerically have been well studied, there is little work to date on robust algorithms for solving poroelasticity models with uncertain inputs and for performing uncertainty quantification (UQ). The Biot model has a number of important physical parameters and inputs whose precise values are often uncertain in real world scenarios. In this work, we introduce and analyse the well-posedness of a new five-field model with uncertain and spatially varying Youngs modulus and hydraulic conductivity field. By working with a properly weighted norm, we establish that the weak solution is stable with respect to variations in key physical parameters, including the Poisson ratio. We then introduce a novel locking-free stochastic Galerkin mixed finite element method that is robust in the incompressible limit. Armed with the `right norm, we construct a parameter-robust preconditioner for the associated discrete systems. Our new method facilitates forward UQ, allowing efficient calculation of statistical quantities of interest and is provably robust with respect to variations in the Poisson ratio, the Biot--Willis constant and the storage coefficient, as well as the discretization parameters.
In this paper we study the linear systems arising from discretized poroelasticity problems. We formulate one block preconditioner for the two-filed Biot model and several preconditioners for the classical three-filed Biot model under the unified relationship framework between well-posedness and preconditioners. By the unified theory, we show all the considered preconditioners are uniformly optimal with respect to material and discretization parameters. Numerical tests demonstrate the robustness of these preconditioners.
Projection-based iterative methods for solving large over-determined linear systems are well-known for their simplicity and computational efficiency. It is also known that the correct choice of a sketching procedure (i.e., preprocessing steps that reduce the dimension of each iteration) can improve the performance of iterative methods in multiple ways, such as, to speed up the convergence of the method by fighting inner correlations of the system, or to reduce the variance incurred by the presence of noise. In the current work, we show that sketching can also help us to get better theoretical guarantees for the projection-based methods. Specifically, we use good properties of Gaussian sketching to prove an accelerated convergence rate of the sketched relaxation (also known as Motzkins) method. The new estimates hold for linear systems of arbitrary structure. We also provide numerical experiments in support of our theoretical analysis of the sketched relaxation method.
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