# A scalable preconditioning framework for stabilized contact mechanics with hydraulically active fractures

A preconditioning framework for the coupled problem of frictional contact mechanics and fluid flow in the fracture network is presented. The porous medium is discretized using low-order continuous finite elements, with cell-centered Lagrange multipliers and pressure unknowns used to impose the constraints and solve the fluid flow in the fractures, respectively. This formulation does not require any interpolation between different fields, but is not uniformly inf-sup stable and requires a stabilization. For the resulting 3 x 3 block Jacobian matrix, we design scalable preconditioning strategies, based on the physically-informed block partitioning of the unknowns and state-of-the-art multigrid preconditioners. The key idea is to restrict the system to a single-physics problem, approximately solve it by an inner algebraic multigrid approach, and finally prolong it back to the fully-coupled problem. Two different techniques are presented, analyzed and compared by changing the ordering of the restrictions. Numerical results illustrate the algorithmic scalability, the impact of the relative number of fracture-based unknowns, and the performance on a real-world problem.

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01/29/2020

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## 1 Introduction

In recent years, attention has grown around novel technologies and applications in the subsurface, like geothermal energy production pan2019establishment ; wei2019numerical ; asai2019efficient , hydraulic fracturing williams2019discursive ; tan2019politics ; krzaczek2020simulations , CO sequestration fan2019thermo ; li2019coupled ; liu2019tutorial and underground gas storage zhou2019seismological ; karev2019geomechanical ; firme2019salt . In these contexts, one of the key components is the simultaneous simulation of frictional contact mechanics and fluid flow in faults and fractures, which represent tightly coupled physical processes. In fact, the aperture and slippage between the contact surfaces drive the fluid flow in the fractures, while the pressure variation perturbs the stress state in the surrounding medium and influences the contact mechanics itself. To achieve the desired accuracy, large domains are usually required, with high resolution representations of geological structures and their heterogeneous properties fergamjantea10 ; castelletto2013geological , and, specifically, of faults and fracture networks zoback2010reservoir ; goodman1968model ; ferronato2008numerical ; GarKarTch16 ; Set_etal17 ; shakiba2015using ; ren2016fully ; wong2019investigation ; wu2019integrating ; deb2009extended ; zhang2011extended ; mohammadi2012xfem ; flemisch2016review ; Berrone2017768 ; vahab2017numerical ; khoei2018enriched ; Berrone2019C317 ; Berrone2021B381 . It is, therefore, natural to have a growing demand towards the development of sophisticated models of increasing size, which are computationally intensive and require better and better performances. A key factor in this sense is the linear solver, which is usually by far the most time-consuming component in a real-world simulation koric2016sparse ; franceschini2019robust .

In this work, we analyze the simulation of frictional contact mechanics coupled with the fluid flow in a fracture network and present a scalable and efficient preconditioning framework for the linear system arising from the discretization and linearization of the coupled problem. As to the discretization approach, we elect to use the Discrete Fracture Model (DFM) GarKarTch16 , i.e., an explicit representation of the fracture surfaces, while the constraints are imposed with the aid of Lagrange multipliers hild2010stabilized ; JhaJua14 ; FraFerJanTea16 ; berge2020finite ; koppel2019stabilized . As it is common in geological and reservoir simulations, we rely on low-order finite elements for the mechanics and a cell-centered finite volume scheme for the fluid flow. Lagrange multipliers are the contact forces acting on the fracture surfaces as a cell-centered variable, thus sharing the same representation as the fluid pressure field with no interpolation needed. The details of this discretization scheme are described in fr2020alg . This approach is unstable in the Ladyzhenskaya-Babuška-Brezzi (LBB) sense, i.e., it does not uniformly satisfy the inf-sup condition (wohlmuth2011variationally, , Section 3.1), and requires a stabilization. In this work, we use the global algebraic approach introduced in the reference work fr2020alg . The Jacobian matrix arising from the described problem is non-symmetric with a block structure, which has to be properly preconditioned to allow for a robust, scalable and efficient solution with the aid of Krylov subspace solvers.

It is well known that iterative methods based on projections/orthogonalizations onto Krylov subspaces saad2003iterative are in practice mandatory to solve large and sparse linear systems deriving from the discretization of PDEs, because they allow for a lower complexity, smaller memory requirement, and better degree of algorithmic parallelism than direct methods davis2006direct . However, robustness, scalability and computational efficiency of this class of methods is tightly connected with the choice of a proper preconditioning technique saad2003iterative

. Roughly speaking, preconditioners are approximate applications of the system matrix inverse, and, from the algebraic viewpoint, can be classified into three main categories: (i) incomplete factorizations

saad1994ilut ; lin1999incomplete ; benzi2002preconditioning , (ii) approximate inverses benzi1996sparse ; tang1999toward ; huckle2003factorized ; janfergam10 ; janfer11 ; janna2015fsaipack , and (iii) multilevel methods, i.e., domain decomposition janfergam13 ; dolean2015introduction ; zampini2016pcbddc ; badia2016multilevel ; li2017low and multigrid-like techniques mccormick1982multigrid ; stuben1983algebraic ; brandt1986algebraic ; stuben2001review ; notay2012aggregation ; brezina2005adaptive ; vanvek1996algebraic ; brezina2006adaptive ; brandt2011bootstrap ; brandt2014bootstrap ; Pasetto20171159 ; dambra2018bootcmatch ; dambra2019improving ; paludetto2019novel . A key feature for a modern preconditioning framework is the algorithmic scalability, i.e., the ability to solve an increasingly refined problem with an approximately constant number of iterations of the Krylov solver. This property is particularly important in view of the development of problems of increasing size by exploiting the availability of massively parallel computational platforms. Incomplete factorizations and approximate inverses can exhibit amazing performances, but do not have a linear complexity with the system size. By distinction, multilevel methods can have a lower performance on a single system, but are designed to be optimal with respect to the scalability issue. Algebraic multigrid (AMG, xu2017algebraic ) is one of the most effective multilevel approaches and consists of the complementary use of: (i) a smoother that reduces high frequency errors, (ii) a coarse grid correction that reduces low frequency errors, and (iii) restriction and interpolation operators, to move from one grid to another. Starting from the original works, e.g., ruge1987algebraic , a wide range of multigrid approaches has appeared in the literature, extending the applicability of this method, originally designed for elliptic PDEs, to both non-symmetric manteuffel2018nonsymmetric ; manteuffel2019nonsymmetric and block matrices webster2016stabilisation ; brenner2014multigrid ; chen2015multigrid ; brenner2018multigrid ; wiesner2021algebraic ; brenner2020multigrid . Nonetheless, robustness and efficiency is still an open issue for AMG whenever used as a black-box tool in problems with these algebraic properties. The Jacobian matrix arising from the model considered herein is a non-symmetric block matrix and, despite the available studies for similar problems, none of them can be straightforwardly and effectively applied to our case. In the context of geomechanical simulations, only a few studies on block Jacobian systems aagaard2013domain ; franceschini2019block ; wiesner2021algebraic are found by the authors.

The purpose of this work is to design a scalable preconditioning framework for the block matrix arising from the coupled simulation of frictional contact mechanics and fluid flow in the fracture network. The idea is to exploit the inherent physics-based block subdivision and the scalability of AMG techniques available from the literature. The full system is first restricted to a single-physics problem, then approximately solved by AMG, and finally prolonged back to the original size. According to the selected restriction ordering, different approaches can be derived. In this work, we consider two different options and investigate advantages and drawbacks in order to find the most appropriate algorithm for real-world simulations. The paper is organized as follows. Section 2 introduces the physical problem in both the strong and weak forms, in order to understand the meaning and features of each block of the Jacobian system. In Section 3, the preconditioning framework is presented, with a detailed analysis of two selected options. Finally, Section 4 presents a set of numerical results with the aim of comparing the proposed approaches and investigating the algorithmic scalability in both theoretical and real-world benchmarks. A few concluding remarks close the paper.

## 2 Problem statement

We model the deformation of an open elastic domain , assuming quasi-static conditions and infinitesimal strains within the open time interval . We denote by its boundary, with , and

the outer normal vector to

, while a set of internal boundaries represents a fracture network consisting of surfaces. The external boundary is subdivided into two non-overlapping subsets, and , where Dirichlet and Neumann boundary conditions apply, respectively. Each fracture consists of two overlapping surfaces, and , with the orientation defined by a unitary vector orthogonal to the fracture plane. By convention, we choose . The pressure field is defined on the union of the two-dimensional (2D) domains , with a one-dimensional (1D) curve defining the boundary of each fracture and . The curve is subdivided into two non-overlapping subsets, and , where Dirichlet and Neumann boundary conditions for the pressure field are imposed. The vector denotes the outer normal direction to

. The fluid is assumed to be incompressible, and body forces and buoyancy effects are neglected. The projection of the stress tensor

along , , is the traction vector over , with and its normal and tangential component, respectively, with respect to the fracture-local reference frame. The traction on controls the possible slipping and aperture of the fracture according to the Coulomb frictional law. A schematic representation of the considered conceptual framework is shown in Figure 1.

The strong form of the initial boundary value problem (IBVP) can be stated as follows KikOde88 ; Lau03 ; Wri06 ; fr2020alg : given the fluid discharge , the prescribed boundary displacement and traction , the prescribed fracture boundary pressure and flux , the initial displacement and pressure , find the displacement , the traction , and pressure such that:

 −∇⋅σ(u) =0 in Ω×T (linear momentum balance), (1a) ˙gN(u)+∇⋅q(u,p) =qs in ∪nfi=1Γi×T (mass balance), (1b) t−pni =0 on ∪nfi=1Γi×T (traction balance), (1c) respecting the boundary conditions u =¯u on ∂Ωu×T (prescribed boundary displacement), (1d) σ(u)⋅nΩ =¯t on ∂Ωσ×T (prescribed boundary traction), (1e) p =¯p on ∪nfi=1∂Γi,p×T (prescribed boundary pressure), (1f) q(u,p)⋅mi =¯q on ∪nfi=1∂Γi,q×T (prescribed boundary flux), (1g) and initial conditions u|t=0 =u0 in ¯¯¯¯Ω (initial displacement), (1h) p|t=0 =p0 in ∪nfi=1¯¯¯¯Γi (initial pressure), (1i) subject to the constraints over each Γi and for every time in T tN=t⋅ni ≤0, gN=⟦u⟧⋅ni≥0,tNgN =0 (normal contact conditions), (1j) ∥tT∥2−τmax(tN) ≤0, ˙gT⋅tT−τmax(tN)∥˙gT∥2 =0 (Coulomb frictional law). (1k)

In the problem statement, is the Cauchy stress tensor, with C the fourth-order elasticity tensor; is the fluid volumetric flux in the fracture domain according to Darcy’s law witherspoon1980validity —assuming laminar flow—with the fluid pressure gradient, the fluid viscosity (constant), and the isotropic fracture hydraulic conductivity modeled as in GarKarTch16 :

 Cf=Cf,0+g3N12, (2)

with the conductivity related to two irregular surfaces that are in contact kamenov2013laboratory ; denotes the relative displacement across , where and are the normal and tangential components, respectively, and and are the restrictions of on and ; is the limit value provided by the static Coulomb criterion, with and the cohesion and friction angle, respectively. Since we employ a static Coulomb criterion, the tangential velocity in (1k) is replaced with the tangential displacement increment wohlmuth2011variationally with respect to the previously converged time-step.

In our framework, we assume to be fixed with no propagation. The domain is partitioned into three portions, where the following contact conditions occur:

• stick on : the fracture is closed () and the traction vector is unknown;

• slip on : the fracture is closed in the normal direction ( and is unknown), but a slip displacement between and is allowed for, with ;

• open on : the fracture is fully open and a free relative displacement is allowed for, with .

For additional details regarding the governing formulation, we refer the reader to KikOde88 ; Lau03 ; Wri06 ; fr2020alg .

### 2.1 Discrete weak form

In the solution to the model problem (1), the traction used as a primary variable plays the role of Lagrange multipliers. Denoting with the appropriate -inner product of scalar, vector or tensor functions in the spatial domain , we introduce the finite-dimensional subspaces , and :

 Vh⊂V ={η∈[H1(Ω)]3:η=¯¯¯¯u on ∂Ωu}, (3a) Mh(thN)⊂M(tN) ={μ∈[L2(Γ)]3:μN≤0,(μ,v)Γ≤(τmax(tN),||vT||2)Γ,v∈[H1/2(Γ)]3] with vN≥0}, (3b) Ph⊂P ={χ∈L2(Γ)}, (3c)

and the discrete approximations of :

 uh=nu∑i=1ηiui∈Vh,th=nt∑j=1μjtj∈Mh(thN),ph=np∑k=1χkpk∈Ph, (4)

where, as before, the pedices and denote the components of a vector function along the normal and tangential direction with respect to a fracture-local reference frame on every . In (4), , , and denote the number of discrete displacement, traction and pressure unknowns. The weak form of (1) reads fr2020alg : find such that

 Ru =(∇sη,σ)Ω+nf∑i=1(⟦η⟧,th−phni)Γi−(η,¯t)∂Ωσ=0 ∀η∈Vh0, (5a) Rt,i =(thN−μN,gN)Γi+(thT−μT,ΔgT)Γi≥0 ∀μ∈Mh(thN), i=1,…,nf, (5b) Rp,i =(χ,ΔgNΔt)Γi+[χ,ph]Fi−FFi(χ)+GFi(χ)−(χ,qs)Γi=0 ∀χ∈Ph, i=1,…,nf, (5c)

where is with homogeneous conditions along , is the time step size, and is a weighted inner product representing the classical two-point flux approximation (TPFA) scheme. This is introduced to allow a unified presentation of the coupled finite element/finite volume model EymGalHer00 ; EymGalHer07 ; Age_etal10 . In particular, we have:

 [χ,ph]F=∑Eint(χ|φL−χ|φK)ΥKL(ph|φL−ph|φK)+∑Epχ|φKΥKph|φK, (6)

where and represent the set of edges included in and , respectively; and are the two adjacent cells and ; and is the harmonic average of one-sided transmissibility and associated to and . Finally, and collect the boundary conditions to be prescribed on and , respectively. For further details, we refer the reader to fr2020alg .

To solve the problem (5), we transform the variational inequality (5b) into a variational equality. For this purpose, we apply an active-set algorithm, as described in nocedal2006numerical ; antil2018frontiers ; fr2020alg , which allows to identify the subdivision into stick/slip/open regions for every . At a given step of the active-set algorithm, the stick/slip/open regions of each fracture are fixed and the inequality (5b) becomes:

 Rt,i=(μ,g)Γsticki+(μN,gN)Γslipi+1k(μT,thT−t∗T)Γslipi+1k(μ,th)Γopeni=0,i=1,…,nf, (7)

with a coefficient needed to ensure the dimensional consistency of the equation. Introducing in (5a), (7) and (5c) the finite-dimensional bases of , and yields the following system of nonlinear discrete residual equations:

 (8)

which is solved by a Newton-Krylov method. In (8), the algebraic vectors , and collect the coefficients , and of the discrete displacement, traction and pressure fields in (4) and is the active-set counter. After convergence of the Newton-Krylov method at the -th step of the active-set algorithm, a consistency check is carried out in order to verify whether the assumed stick/slip/open region subdivision meets the Coulomb frictional conditions. If not, the region subdivision is updated and a new step is performed. The algorithm stops when the consistency check does not require to modify the stick/slip/open region subdivision. At this point, convergence is achieved and the solution is sought at the following time step.

The finite element/finite volume spaces used in this work are the same as in fr2020alg , i.e., first-order continuous finite elements for displacements and face-centered piecewise-constant elements for tractions and pressures, as schematically represented in Figure 1. To model the fractures, we use a DFM approach with a conforming mesh GarKarTch16 , hence any is represented by a set of finite element faces. Thus, displacement unknowns are located on mesh vertices, while traction and pressure unknowns are on fracture faces (Figure 1), with equal to three times the number of 3D finite element nodes, equal to the number of 2D faces discretizing the fracture network, and equal to . Displacements are represented in the global reference system and tractions are represented in a face-based local reference frame. This approach is intrinsically unstable, as it does not fulfill the inf-sup condition wohlmuth2011variationally . In this work, we use the global algebraic stabilization proposed in fr2020alg , which relaxes the zero jump and the impenetrability conditions between the two fracture surfaces in the traction balance equation, and the fluid incompressibility constraint in the mass balance equation. Only stick and slip portions are involved in the traction balance, being the tractions in the open part known. With the introduction of the stabilization, equations (7) and (5c) become:

 Rt,i i=1,…,nf, (9a) Rp,i =(χ,ΔgNΔt)Γi+[χ,ph]Fi−FFi(χ)+GFi(χ)−(χ,qs)Γi+1Δtjp(χ,ph)Γi=0, i=1,…,nf, (9b)

where and are the stabilizing bilinear forms for the traction and pressure field, respectively. In particular, we have

 jt(μ,th)=∑ϵ∈Eint1|ϵ|∫ϵ⟦μ⟧ϵ⋅Υ(ϵ)⋅⟦th⟧ϵdl, (10)

with denoting the jump of a quantity across the generic internal edge and is a positive definite second-order tensor providing the appropriate scaling. The discrete formulation of is fully provided in fr2020alg . The contribution is computed as the normal projection of with respect to the surface .

At a given active-set iteration , the Newton linearization of (8), which now includes also the stabilization terms, generates a sequence of linear systems and vector updates. To advance by one Newton iteration , we have to:

 solveJℓ,(k)δx=−rℓ,(k) ⇒⎡⎢⎣AC1Q1C2−H0Q20T⎤⎥⎦ℓ,(k)⎡⎢⎣δuδtδp⎤⎥⎦=−⎡⎢⎣rurtrp⎤⎥⎦ℓ,(k), (11) updatexℓ,(k+1)=xℓ,(k)+δx ⇒⎡⎢⎣utp⎤⎥⎦ℓ,(k+1)=⎡⎢⎣utp⎤⎥⎦ℓ,(k)+⎡⎢⎣δuδtδp⎤⎥⎦.

The submatrices in the block Jacobian read:

 [A]ij =(∇sηi,C:∇sηj)Ω, i=1,nu, j=1,nu, (12a) ij =(⟦ηi⟧,μj)Γ, i=1,nu, j=1,nt, (12b) ij =−(⟦ηi,N⟧,χj)Γ, i=1,nu, j=1,np, (12c) ij =(μi,⟦ηj⟧)Γℓ,stick+(μi,N,⟦ηj,N⟧)Γℓ,slip−1k(μi,T,(∂t∗T∂ΔgT)∣∣∣ℓ,(k)⋅⟦ηj,T⟧)Γℓ,slip, i=1,nt, j=1,nu, (12d) ij =−1k(μi,T,μj,T)Γℓ,slip+1k(μi,T,(∂t∗T∂tN)∣∣∣ℓ,(k)μj,N)Γℓ,slip−1k(μi,μj)Γℓ,open+jt(μi,μj)Γℓ,%stick∪Γℓ,slip, i=1,nt, j=1,nt, (12e) ij =1Δt(χi,⟦ηj,N⟧)Γℓ,open+∂([χi,ph]F)∂uj∣∣ ∣∣ℓ,(k)−∂(FF(χi))∂uj∣∣∣ℓ,(k), i=1,np, j=1,nu, (12f) ij =∂([χi,ph]F)∂pj∣∣ ∣∣ℓ,(k)+1Δtjp(χi,χj)Γℓ, i=1,np, j=1,np. (12g)

The partial derivatives appearing in (12) are reported in (fr2020alg, , Appendix A).

### 2.2 Linear system

We focus our attention on the linear system solution and the design of robust, scalable and efficient preconditioners for the block matrix of equation (11). The global matrix is large, sparse, and non-symmetric, with properties that change with the evolution of the stick/slip/open regions in the fracture network. A representative evolution of the non-zero pattern of during a full simulation is shown in Figure 2. The features that follow are worth summarizing.

1. The first block row of includes the contributions arising from the linear momentum balance of the 3D domain . All the submatrices do not depend on the fracture state and can be assembled once at the beginning of the whole simulation if an elastic constitutive law is used. In particular, is the classical symmetric positive definite (SPD) elastic stiffness matrix, while and are tall rectangular blocks collecting a surface measure of the fracture elements and transferring tractions and pressures to the 3D body as applied forces.

2. In the second block row of , varies as the stick/slip/open fracture regions evolve through the active-set algorithm, in both the entry values and the non-zero pattern (Figure 2). If all the fracture elements are in stick mode, we have that , otherwise the frictional law derivatives appear and .

3. When all fractures belong to the stick region, is the symmetric positive semidefinite (SPSD) stabilization matrix. In case of sliding, non-symmetric diagonal blocks arise, one for each traction component along the local tangential direction to the fracture surface. In the open regions the rows of have a single non-zero entry in the main diagonal, with no contribution from the stabilization term (Figure 2). In any case, is singular and cannot be regularly inverted.

4. The third block row of includes the contributions arising from the fluid mass balance on the fracture network. The coupling between fluid flow and fracture mechanics is controlled by . In particular, when all fracture elements are in stick mode, and is reducible with a symmetric saddle-point matrix as leading block. Otherwise, contributions from the flux derivative with respect to the displacements appear, i.e., entries depend on the current pressure solution (Figure 2). By distinction with and , there is no simple relationship between and , in both the entry values and the non-zero pattern. Denoting with the matrix-to-matrix operator returning a zero row if the corresponding element index belongs to and the original row if the element index belongs to , can be written as:

 Q2=−Ψ(QT1)Δt+Fu, (13)

where collects the contributions from the flux derivatives with respect to the displacements.

5. is the sum of the standard transmissibility matrix arising from the TPFA discretization in the 2D domain and the stabilization contribution. As such, it is SPD with the 5-point stencil of a 2D discrete Laplacian. Moreover, has a block diagonal structure for all non-intersecting fractures. Observe also that traction and pressure fields are always decoupled.

From the observations above, it appears that matrix changes nature with the evolution of the fracture conditions, moving from a reducible matrix with a symmetric saddle-point leading block to a general non-symmetric and indefinite matrix. The objective of our work is to define a unique preconditioning framework ensuring robustness, scalability and computational efficiency for any working situation.

## 3 Preconditioning framework

A preconditioner of is a non-singular operator whose application to a vector resembles as much as possible the action of . The exact application of to some vector provides the vector such that:

 ⎧⎪⎨⎪⎩Avu+C1vt+Q1vp=wuC2vu−Hvt=wtQ2vu+Tvp=wp, (14)

with , , and natural subvectors of , respectively. The objective is to approximate the solution to the multi-physics system (14) by exploiting the physics-based variable partitioning. The system is first reduced to a single-physics problem, and then prolonged back to the full multi-physics space. According to the selected sequence of reductions, different algorithms may arise.

### 3.1 Method no. 1: t-p-u approach

Traction and pressure variables live on the fractures and are mutually decoupled independently on the stick/slip/ open region partitioning. Therefore, it is natural to exploit this condition and perform a simultaneous reduction of both variable sets onto the displacement space. This corresponds to compute and from the second and third equation of (14), respectively, and introduce them in the first equation, thus eliminating both physics from the equilibrium equation. Recall, however, that is singular, so a regular surrogate is needed. A block diagonal approximation can be used instead, where each block is the local stabilization matrix computed for each fracture element. Denoting with such a block-diagonal approximation, we have:

 vt ≃−˜H−1(wt−C2vu), (15a) vp =T−1(wp−Q2vu). (15b)

With (15a) and (15b), the first equation of (14) becomes:

 (A+C1˜H−1C2−Q1T−1Q2)vu≃wu+C1˜H−1wt−Q1T−1wp, (16)

which is a single-physics equilibrium equation on the 3D domain where the elimination of fracture tractions and pressures introduces fictitious stiffness contributions. The matrix at the left-hand side of (16) is the Schur complement :

 S=A+C1˜H−1C2−Q1T−1Q2. (17)

Solution to (16) provides , which, introduced into equations (15), yields the final vector . The multi-physics reduction order performed in this case is traction-pressure-displacement (t-p-u) and is schematically summarized in Figure 3.

The computation and inversion of in (17) cannot be performed exactly. The Schur complement is explicitly approximated by :

 S≃˜S=A+C1˜H−1C2−Q1˜T−1Q2, (18)

where is a diagonal surrogate for . The inverse can be applied inexactly by means of an AMG operator, which can be efficiently used in mechanical problems preserving a linear complexity with respect to the problem size. This is a key property to guarantee the solver scalability. Recent examples of effective AMG preconditioners are, for instance, taken from the References brandt2014bootstrap ; dambra2018bootcmatch ; dambra2019improving ; paludetto2019novel . In this work, we use an aggregation-based multigrid as the reference AMG operator. Specifically, the application of is approximated by GAMG may2016extreme , the state-of-the-art aggregation based multigrid provided by the PETSc package petsc-user-ref .

The construction and application of the resulting preconditioning operator with the t-p-u approach is summarized in Algorithms 1 and 2. The operator gives a matrix with the diagonal blocks of , while the operator applies the selected AMG preconditioner of to the vector . Since is generally much larger than and , the cost for applying the exact inverse of and is negligible with respect to the AMG algorithm for . Hence, the latter can be roughly assumed as the cost per iteration for the application.

From an algebraic viewpoint, the preconditioning operator arising from the t-p-u approach can be written as an inexact block LDU factorization of . Using the permutation matrix :

 Q1=⎡⎢⎣0It000IpIu00⎤⎥⎦, (19)

where

is the identity matrix in

and

the zero matrix of proper size, the block LDU factorization reads:

 Q1JQT1≃L1D1U1, (20)

with:

 L1 =⎡⎢ ⎢⎣It000Ip0−C1˜H−1Q1T−1Iu⎤⎥ ⎥⎦, D1 =⎡⎢⎣−˜H000T000˜S⎤⎥⎦ and U1 =⎡⎢ ⎢⎣It0−˜H−1C20IpT−1Q200Iu⎤⎥ ⎥⎦. (21)

Hence, the final algebraic expression of is:

 M−11=⎡⎢ ⎢⎣It0˜H−1C20Ip−T−1Q200Iu⎤⎥ ⎥⎦⎡⎢ ⎢⎣−˜H−1000T−1000˜S−1⎤⎥ ⎥⎦⎡⎢ ⎢⎣It000Ip0C1˜H−1−Q1T−1Iu⎤⎥ ⎥⎦. (22)
###### Remark 3.1

The multi-physics reduction approach proposed herein can be equivalently recast in other ways as well. Since we use a twofold approximation for , i.e., exact in , and inexact in , can be regarded as a member of the mixed constraint preconditioner class Bergamaschi2008 ; ferjangam08 ; Ferronato2010 . Similarly, the upper and lower block triangular factors in (22) play the role of decoupling operators for the original multi-physics problem and are the outcome of the general-purpose algebraic procedure defined in Ferronato2019 . Finally, can be also regarded as an example of application in a block non-symmetric context of the multigrid reduction framework Bui2018 ; Bui2020 , where fracture and body variables play the role of fine and coarse nodes, respectively, and replaces in matrix .

Let us introduce the matrices:

 EH =It−˜H−1H, (23a) ES =Iu−˜S−1S, (23b)

which can be regarded as a matrix measure of the quality of the approximations and introduced in . The following result holds.

###### Proposition 3.2.

The eigenvalues

of the preconditioned matrix with the t-p-u approach are either 1, with multiplicity , or such that:

 |λ1−1|≤(1+ζ)max{εH,εS}, (24)

with , , and

 Z=[˜H−1C2˜S−1C1−˜H−1C2˜S−1C10] (25)

for any compatible matrix norm.

###### Proof.

Recalling equations (19) and (22) and introducing the error matrices (23), the preconditioned matrix with the t-p-u approach reads:

 (26)

which has unitary eigenvalues. The remaining eigenvalues are those of the matrix obtained by dropping the second block row and column from (26):

 H=It+u+(Z−It+u)E1, (27)

with the identity matrix of order and :

 E1=[EH00ES]. (28)

The eigenvalues of satisfy the bound (24), thus closing the proof. ∎

###### Remark 3.3

Proposition 3.2 shows that the distance of from the identity and the approximation quality of are key factors for the overall performance of . While is fixed, notice, however, that of equation (18) has algebraic properties that change with the fracture state and the evolution of the stick/slip/open regions throughout the active-set algorithm. In stick mode, the contribution is symmetric positive semidefinite and . Hence, is SPD. Also in slip mode the contribution is positive definite and , so remains positive definite, though slightly non-symmetric. With open elements, however, depends on the current pressure solution and no theoretical considerations can be made in general. In these conditions, is an indefinite non-symmetric matrix.

### 3.2 Method no. 2: t-u-p approach

An alternative multi-physics reduction sequence relies on the scheme sketched in the rightmost panel of Figure 3. Introducing the traction variables (15a) into the first equation of system (14) yields:

 (A+C1˜H−1C2)vu+Q1vp≃wu+C1˜H−1wt, (29)

where the matrix:

 S1=A+C1˜H−1C2 (30)

is the first-level Schur complement. From a physical viewpoint, is an elasticity matrix with fictitious stiffness contributions arising along the fractures from the traction elimination. Then, a second reduction is needed by computing from (29) and introducing it in the third equation of (14):

 (T−Q2S−11Q1)vp≃wp−Q2S−11(wu+C1˜H−1wt). (31)

The matrix at the left-hand side of equation (31) is the second-level Schur complement:

 S2=T−Q2S−11Q1, (32)

which, from a physical point of view, represents a modified transmissibility matrix including the effect of the stiffness of the 3D medium surrounding the fractures. Hence, the multi-physics reduction order is traction-displacement-pressure (t-u-p).

The computation of can be performed exactly, but its inverse has to be approximated. Since its nature is the same as that of of equation (17), we can effectively use an AMG operator, such as GAMG. We denote with the operator that approximately applies . By distinction, cannot be computed exactly. Recalling the physical interpretation of and , we can approximate the contribution