Robust block preconditioners for poroelasticity☆

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Highlights

  • •

    Show the well-posedness of Biot model in different norms under a unified framework.

  • •

    Design several preconditioners for the Biot model

  • •

    Analyze and prove the robustness of the designed preconditioners.

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    Numerical results are presented to verify the theoretical results.

Abstract

In this paper we develop and analyze several preconditioners for the linear systems arising from discretized poroelasticity problems. The preconditioners include one block preconditioner for the two-field Biot model and several preconditioners for the classical three-field Biot model. We manage to analyze these different preconditioners under a same theoretical framework and show that all of them are uniformly optimal with respect to material and discretization parameters. Numerical tests demonstrate the robustness of these preconditioners.

Introduction

Poroelasticity, the study of the fluid flow in porous and elastic media, couples the elastic deformation with the fluid flow in porous media. As one popular poroelasticity model, the Biot model has wide applications in geoscience, biomechanics, and many other fields. There are many issues needed to be addressed in numerical simulations of poroelasticity such as the numerical instability of pressure variable under certain conditions [1], [2], [3], [4]. One source of this instability is the instability of the finite element approximation for the coupled systems [2], [3]. This motivates us to study the well-posedness of the finite element discretization.

Another interesting topic is the development of efficient linear solvers. Direct solvers have poor performance when the size of problems becomes large. Iterative solvers are good alternatives, as they exhibit better scalability, but the convergence of iterative solvers is known to be much problem-dependent such that there is a need for developing parameter-robust preconditioners. For example, the multigrid preconditioned Krylov subspace method usually has optimal convergence rate for the Poisson equation and many other symmetric positive definite problems [5], [6]. However, for poroelasticity problems, coupled systems of equations must be solved, which are known to be indefinite and ill-conditioned [7]. Preconditioning techniques for poroelasticity problems have been the subject of considerable research in the literature [3], [8], [9], [10], [11], [12], [13], [14] and most of the techniques developed are based on the Schur complement approach. In [10], [11], diagonal approximation of the Schur complement preconditioner is used to precondition two-field formulation of the Biot model. In [12], [13], Schur complement preconditioners are also studied for two-field formulation with the algebraic multigrid (AMG) as the preconditioner for the elasticity block. In [3], Schur complement approaches for three-field formulation are investigated. Recently, robust block diagonal and block triangular preconditioners are developed in [15] for two-field Biot model. And for classical three-field Biot model, the robust block preconditioners are designed in [16], [17] based on the uniform stability estimates. Robust preconditioner for a new three-field formulation introducing a total pressure as the third unknown is analyzed in [18]. Robust block diagonal and block triangular preconditioners are also developed in [19] based on the discretization proposed in [20]. Other robustness analysis for fixed-stress splitting method and Uzawa-type method for multiple-permeability poroelasticity systems are presented in [21] and [22].

The focus of this paper is on the stability of the linear systems after time discretization and several robust preconditioners for the iterative solvers under the unified relationship framework between well-posedness and preconditioners. The block preconditioners in [15] for two field formulation and in [16], [19] for the three field formulation can be briefly written in this framework. In addition, we analyze the well-posedness of the linear systems and propose other optimal preconditioners for the Biot model [2] based on the mapping property [23]. By proposing optimal block preconditioners, we convert the solution of complicated coupled system into that of a few symmetric positive definite (SPD) systems on each of the fields.

The rest of this paper is organized as follows. In Section 2, we give a brief introduction of the Biot model. In Section 3, we introduce one theorem in order to prove well-posedness. In Section 4, we address the unified framework indicating the relationship between preconditioning and well-posedness of linear systems. In Section 5 and Section 6, we show the well-posedness and several optimal preconditioners for the Biot model under the unified framework. In Section 7, we present numerical examples to demonstrate the robustness of these preconditioners.

Section snippets

The Biot model

The poroelastic phenomenon is usually characterized by the Biot model [24], [25], which couples structure displacement u, fluid flux v, and fluid pressure p. Consider a bounded and simply connected Lipschitz domain Ω⊂Rn(n=2,3) of poroelastic material. As the deformation is assumed to be small, we assume that the deformed configuration coincides with the undeformed reference configuration. Let σ denote the total stress in this material. From the balance of the forces, we first have −∇⋅σ=f, in Ω.

Well-posedness of linear systems

In this section, we introduce the theorem to prove the well-posedness of the following saddle point problem: Find (u,p)∈M×N such that ∀(ϕ,q)∈M×N, the following equations hold a(u,ϕ)+b(ϕ,p)=〈f,ϕ〉,b(u,q)−c(p,q)=〈g,q〉.Here, M and N are given Hilbert spaces with the inner products (⋅,⋅)M and (⋅,⋅)N, respectively. The corresponding norms are denoted by ‖⋅‖M and ‖⋅‖N.

Given b(⋅,⋅), the following kernel spaces are important in the analysis: Z={u∈M|b(u,q)=0,∀q∈N},K={p∈N|b(ϕ,p)=0,∀ϕ∈M}.

We consider the

Relationship between preconditioning and well-posedness

Given that a variational problem is well-posed, an optimal preconditioner can be developed, in order to speed up Krylov subspace methods, such as Conjugate Gradient Method (CG) and Minimal Residual Method (MINRES). In order to illustrate this fact, we first consider the following variational problem:

Find x∈X, such that L(x,y)=〈f,y〉,∀y∈X,where X is a given Hilbert space and f∈X′.

The well-posedness of the variational problem (14) refers to the existence, uniqueness, and the stability ‖x‖X≲‖f‖X′

A two-field formulation

The preconditioning for the two-field system (6) has been studied extensively in the literature [10], [11], [12], [13], where the Schur complement approach is usually used to develop preconditioners. In this paper, similar to [15], we briefly formulate a preconditioner based on the well-posedness of the linear systems for the two-field Biot model.

We first study the well-posedness of (6), beginning by changing the variable p̃=−αp in order to symmetrize (6). With an abuse of notation, we still

A three-field formulation

In this section, we will show the well-posedness of the three field formulation (5), briefly formulate the diagonal block robust preconditioners of [16], [19] as special cases, and propose some new preconditioners for the three field formulation guided by the well-posedness.

Numerical tests

In 2D case, we test the preconditioners using the poroelastic footing experiment (see [4]). The domain is Ω=(−4,4)×(−4,4). Define Γ1={(x,y)∈∂Ω,|x|≤0.8,y=4},Γ2={(x,y)∈∂Ω,|x|>0.8,y=4}.

The boundary conditions are as follows: (σe−pI)n=−104,v⋅n=0,on Γ1,(σe−pI)n=0,p=0,on Γ2,u=0,v⋅n=0,on ∂Ω∕(Γ1∪Γ2).We assume that the fluid storage coefficient is S=0 and the other material parameters are varying in huge range.

We discretize the problem using FEniCS [42]. We show the robustness of the preconditioners

Concluding remarks

In this paper we study the well-posedness of the linear systems arising from discretized poroelasticity problems. We formulate block preconditioner for the two-field Biot model and several preconditioners for the classical three-field 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. The preconditioners have

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    ☆

    This work of Jinchao Xu is supported by the Computational Materials Sciences Program funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award Number DE-SC0020145.

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