Efficient Preconditioners for Krylov Subspace

Efficient Preconditioners for Krylov Subspace Methods in the Solution of Coupled Consolidation Problems Luca Bergamaschi, 1 2 Massimiliano Ferronato, 3 and Giuseppe Gambolati 4 Abstract A major computational effort in the Finite Element (FE) integration of a coupled consolidation model is the repeated solution in time of the resulting discretized indefinite equations. Because of ill-conditioning, the iterative solution may require the assessment of a suitable preconditioner to guarantee convergence. At each time step, the solution of the linear system Ax = b where K BT A= , is performed by a Krylov subspace method, preconditioned B −C −1 G BT −1 by M = , with G a symmetric positive definite approximaB −C tion of block K. The eigenvalues of the preconditioned matrix are proven to be bounded by the eigenvalues of G−1 K. Moreover, an efficient way to apply the preconditioner to a vector is developed. Numerical results onto test problems of large size reveal that the proposed preconditioner can be more efficient than the standard ILU/ILUT preconditioners.

1 Introduction The time-dependent distribution of displacements and fluid pore pressure in porous media was first mathematically described by Biot [6], who developed the consolidation theory coupling the elastic equilibrium equations with a continuity or mass balance equation to be solved under appropriate boundary and initial flow and loading conditions. A consolidation problem is typically solved in space by a Finite Element (FE) technique giving rise to a system of first-order differential equations, the solution to which is usually addressed by an appropriate time marching scheme. A major computational effort is the repeated solution in time of the resulting discretized indefinite equations. In particular, with the small time integration steps typically required in the early phase 1

Corresponding author Department of Mathematical Methods and Models for Scientific Applications, University of Padua, Italy, [email protected] 2 Department of Mathematical Methods and Models for Scientific Applications, University of Padua, Italy, [email protected] 2 Department of Mathematical Methods and Models for Scientific Applications, University of Padua, Italy, [email protected] 2

1

of the analysis the final linear system may be severely ill-conditioned [11], so that obtaining an accurate solution may prove difficult by any numerical approach. Because of the large size of realistic three-dimensional (3-D) consolidation models (and particularly so in problems related to fluid withdrawal/injection from/into geological formations) the use of iterative solvers is generally recommended. Among them, projection (or conjugate gradient-like) methods based on Krylov subspaces for unsymmetric indefinite systems, such as Bi-CGSTAB (Bi-Conjugate Gradient Stabilized [23]), are attracting a growing interest on the grounds of their robustness and efficiency [9, 12, 13, 14, 21, 10]. However, to accelerate convergence a key issue is the selection of an ad hoc efficient preconditioning strategy. The projection solver performance can be also improved by a proper preliminary left and right scaling [14] which helps stabilize CG-like methods preconditioned by an incomplete triangular factorization. The aim of the present paper is to investigate the performance and the robustness of a novel block preconditioner developed for the solution to the symmetric indefinite system of equations arising from the FE integration of the coupled consolidation equations. The outcome from two realistic medium and large size 3-D problems is compared to the solution strategy suggested by Gambolati et al. [14], which has proved to be, at present, one of the most effective solution techniques for large size, possibly ill-conditioned, FE coupled consolidation models. In particular, the comparison is performed for both normally conditioned and severely ill-conditioned problems. Finally, some remarks on the potential of the proposed technique close the paper.

2 Finite Element consolidation model The differential equations governing the 3-D coupled consolidation process in fully saturated porous media are derived from the classical Biot’s formulation [6] as modified by van der Knaap [22] and Geertsma [16]: (λ + G)

∂ǫ ∂p + G∇2 ui = α ∂i ∂i

i = x, y, z

∂p ∂ǫ 1 ∇(k∇p) = [φβ + cbr (α − φ)] +α γ ∂t ∂t

(1)

(2)

where cbr and β are the volumetric compressibility of solid grains and water, respectively, φ is the porosity, k is the medium hydraulic conductivity, ǫ the medium volumetric dilatation, α the Biot coefficient, λ and G are the Lam´e constant and the shear modulus of the porous medium, respectively, γ is the specific weight of water, ∇ is the gradient operator, x, y, z are the coordinate directions, t is time, and p and ui are the incremental pore pressure and the components of incremental displacement along the i−direction, respectively. Integration by FE in space yields a system of first order differential equations which 2

can be written as: u f u˙ 0 0 u K −Q = + T fp p˙ Q P p 0 H

(3)

where K, H, P and Q are the elastic stiffness, flow stiffness, flow capacity and flow˙ p} ˙ T are the vectors of the stress coupling matrices, respectively, {u, p}T and {u, unknown variables ui and p and the corresponding time derivatives, and {f u , f p }T is the vector of the nodal loads (f u ) and flow sources (f p ). Eq. (3) can be written in a more compact form as: K1 x + K2 x˙ + f = 0

(4)

where the meaning of the new symbols above is immediately derived from comparison of eqs. (3) and (4). Eq. (4) is integrated in time by the well known θ−method (e.g. [7]): K2 K2 θK1 + (5) − (1 − θ)K1 xm − θf m+1 + (1 − θ)f m xm+1 = ∆t ∆t Eq. (5) constitutes a linear system which has to be solved for the displacement and the pore pressure. The non-symmetric matrix controlling the solution scheme reads:

K2 A = θK1 + ∆t

=

θK QT ∆t

−θQ P θH + ∆t

(6)

Matrix A can be readily symmetrized by multiplying the upper set of equations by 1/θ and the lower set by −∆t, thus obtaining the following sparse 2 × 2 block symmetric indefinite matrix: K B (7) A= B T −C where B = −Q and C = θ∆tH + P . The blocks K and C are both symmetric and positive definite (SPD). In 3-D problems, denoting by n the number of FE nodes, C ∈ Rn×n , B ∈ R3n×n , and K ∈ R3n×3n . The conditioning of matrices (6) or (7) depends on the relationship linking the time step ∆t to the hydro-mechanical properties of the porous medium and the FE discretization. Ferronato et al. [11] have shown that a critical time step ∆tcrit can be defined as: Vγ ∆tcrit = χ(ψ, θ) (8) kE where E is the Young modulus of the porous medium, V the elemental volume, ψ = φβE, and χ is a dimensionless factor depending on ψ, θ, and the element distortion. For ∆t ≤ ∆tcrit the conditioning of A or A suddenly worsens and the solution to system (5) can be quite difficult to get independently on the solver choice. 3

3 Constraint Preconditioner To solve the system Ax = b we employ a Krylov method accelerated with the preconditioner M−1 where G BT . (9) M= B −C In eq. (9) G is an SPD approximation of block K. Its inverse, G−1 , can be viewed as a preconditioner for K, and it is assumed to be explicitly known. The next Theorem states that the proposed preconditioner produces a cluster of the eigenvalues of M−1 A around unity. In particular, there are at least n unit eigenvalues, while the other eigenvalues are bounded by those of G−1 K. Theorem 1. Assume B has maximum rank. Let us denote with αK , βK the smallest and the largest eigenvalues of G−1 K, and assume αK ≤ 1 ≤ βK . Then the eigenvalues λ of M−1 A are either one (with multiplicity at least n) or real positive and bounded by αK ≤ λ < βK

The success of the proposed preconditioner is therefore related to the following conditions: 1. G−1 is a good preconditioner for block K; 2. application of preconditioner M−1 , i.e. solution to the linear system Mx = r, is computationally inexpensive.

3.1 Convergence of the PCG method After some calculation an explicit form for the (right) preconditioned matrix is obtained as: X Y −1 AM = 0 In where the block X is (G + B T C −1 B)−1 (K + B T C −1 B). As is well known (see [17]), the Preconditioned Conjugate Gradient (PCG) method produces a sequence of residuals r k = Pk (AM−1 )r 0 , where Pk denotes a polynomial of degree k. Due rˆ 0 , then r k = to the block structure of the preconditioned matrix, if r 0 = 0 Pk (X k )ˆ r0 , which implies that PCG applied to the preconditioned system be0 haves like PCG applied to the SPD matrix K + B T C −1 B preconditioned with the SPD matrix G + B T C −1 B. It may therefore be concluded that PCG can be successfully applied to our indefinite linear system, as is also proved in [20], provided that 4

the last n components of r 0 are zero. To this aim it is sufficient to start the iterative procedure with x0 = M−1 b. In such case we obtain: rˆ 0 b1 I − X −Y −1 = r 0 = b − AM b = 0 b2 0 0

3.2 Numerical algorithms and implementation In a Krylov-like method at each iteration we have to solve: Mx = r

(10)

where r is the current residual. In the form described above the preconditioner application reads: x1 r1 G BT = (11) x2 r2 B −C Now solving for x1 in the upper set of eqs. (11): x1 = G−1 r 1 − B T x2 and substituting in the lower one gives:

BG−1 B T + C x2 = BG−1 r 1 − r 2

(12)

Matrix S = BG−1 B T + C is the Schur complement matrix of system (10). The efficiency of this procedure rests on: 1. the availability of a good quality and possibly cheap preconditioner G−1 for K; 2. the efficient solution of the linear n × n SPD system (12). The second task can be conveniently fulfilled by the use of a PCG method, thus defining a cycle of inner iterations. However, solving a linear system by PCG at each outer iteration can result in a burden that may slow down significantly the whole algorithm.

3.3 Approximation of the Schur complement matrix It may be easily shown that preconditioner M can be written as the product of three block triangular matrices: In G−1 B T G 0 In 0 M= 0 Im 0 −S BG−1 Im This form is particularly suited to build a compact form of the inverse of M: −1 In 0 G 0 In −G−1 B T −1 M = −BG−1 Im 0 −S −1 0 Im 5

Our aim is now to find an approximation to M−1 that makes its application cheaper. This is accomplished with the incomplete Cholesky factorization of S: S ≈ LLT and the substitution of S above with LLT to give: −1 −1 T I 0 G 0 I −G B n n −1 c = M −BG−1 Im 0 −(LLT )−1 0 Im By the above approach system (12) doesn’t need to be solved, its solution being replaced by the “inexpensive” solution of two triangular and sparse linear systems. c−1 will be referred to in the sequel as inexact constraint precondiPreconditioner M tioner (ICP).

3.4 Choice of matrix G−1 Diagonal approximation of K −1 . A frequent choice of G is diag(K). The preconditioner thus obtained has been successfully used in the context of interior point methods for quadratic optimization (see e.g. [5, 4, 8]) or in the context of numerical integration of PDEs ([21, 10]). However, as stated by Theorem 1, such a preconditioner is as effective as the Jacobi preconditioner applied on K, and in most cases does not lead to solver convergence. Factorized Approximate inverses. Starting from middle ’90s there has been a growing interest in the sparse approximate inverse preconditioners. These preconditioners involve only matrix-vector products. We quote among others SPAI [18], AINV [2, 1] and FSAI (Factorized Sparse Approximate Inverse) preconditioner [19]. The FSAI preconditioners offer a number of advantages over their non factorized counterparts. They preserve the positive definiteness of the problem and provide better approximations to K −1 for the same amount of storage (than non-factorized ones). For an extensive comparative study of sparse approximate inverse preconditioners the reader is referred to [3]. The AINV preconditioner is generally more efficient than FSAI as accelerator of Krylov solvers, due to its flexibility in the generation of the pattern of the approximate inverse factor. The approximation of K −1 provided by AINV is K −1 ≈ G−1 = ZZ T where Z is upper triangular. Then In 0 ZZ T 0 In −G−1 B T −1 c M = −BG−1 Im 0 −(LLT )−1 0 Im ZT 0 Z −ZZ T B T L−T = L−1 BZZ T −L−1 0 L−T = UL 6

(13)

The factorized form (13) of preconditioner M−1 is well suited to implementation. The numerical method described above requires the explicit knowledge of Schur complement matrix S = BG−1 B T + C = BZZ T B T + C. Forming this matrix may be time and memory consuming being the result of two sparse matrix-matrix products and one sparse sum of matrices. However, it should be noted that the evaluation of S0 = BZZ T B T – the main computational burden of S – is independent of timestep and therefore can be done at the beginning of the simulation, once and for all. The complete solution algorithm reads (Algorithm 1):

A LGORITHM 1. ICP-AINV Input: tolAINV , tolILU , tolBiCG • Compute an approximate inverse of K: K −1 ≈ ZZ T with drop tolerance tolAINV . • Compute W = BZ • Compute S0 = W W T • DO i = 1, nstep 1. C = θ∆ti H + P ;

S = S0 + C.

2. Compute an incomplete factorization of S: S ≈ LLT with tolerance tolILU . 3. Solve Ax(i) = b by the Bi-CGSTAB iterative method preconditioned with M−1 . Iteration is completed when the following exit test is satisfied: kr k k ≤ tolBiCG kr 0 k END DO

The actual solution to the linear system at each timestep is accomplished in step 3 of the ICP-AINV algorithm. The classical Bi-CGSTAB iterative method has to be modified in order to provide an efficient application of preconditioner M−1 . An iteration of Bi-CGSTAB algorithm requires twice the application of the preconditioned matrix. c−1 = UL) the As in our case the preconditioner is known in factorized form (i.e. M so called “split” preconditioning technique requires twice the computation of y = LAUx

(14)

In view of (13), we detail in Algorithm 2 the steps needed to compute (14), where v = Ux, (steps 1–4) z = Av, (steps 5 and 6) and y = Lz (steps 6–10). 7

A LGORITHM 2: A PPLICATION OF THE PRECONDITIONED MATRIX 1. solve LT v 1 = x2 2. v ′ = B T v 1 3. w = x1 − Z T v ′ 4. v 2 = Zw 5. z 1 = Kv 1 + B T v 2 6. z 2 = Bv 1 − Cv 2 7. y 1 = Z T z 1 8. w = Zy 1 9. w′ = z 2 − Bw 10. Solve Ly 2 = w′

4 Test problem

11 00 00 11 00 11 00 11 00 11 00 11

1 0 0 1 0 1 0 1 0 1 0 1

p=0

aquifer

100 m 200 m 100 m

p=0

3000 m

1000 m

A vertical cross-section of the cylindrical porous volume used as test problem is shown in Figure 1. The medium consists of a sequence of alternating sandy and clayed layers, with the hydraulic conductivity ksand = 10−5 m/s and kclay = 10−8 m/s, the Poisson ratio ν = 0.25, and the Young modulus E = 833.33 MPa, corresponding to a uniaxial vertical compressibility cM = 10−3 MPa−1 . Standard Dirichlet conditions are prescribed, with fixed outer and bottom boundaries, and zero pore pressure variation on the top and outer surfaces (see Figure 1). The second order Crank-Nicolson finite difference scheme is used (θ = 0.5), with a variable time step ∆t. The sample problem is solved using fully 3-D grids. In the first test case, denoted as M3Dsm, the medium is discretized into tetrahedral elements by projecting a plane

11 00 00 11 00 11 00 11 00 11 11 0011 00 000 111 0011 11 00 000 111

4500 m

1 0 0 1 0 1 0 1 0 00011 111 0011 001 00011 111 0011 00

sand clay

Figure 1: Schematic representation of a vertical cross-section of the stratified porous medium used as test problem. 8

triangulation made of 209 nodes and 400 triangles onto 17 layers located at different depths [15]. The grid M3Dsm totals n = 3553 nodes with a global matrix size N equal to 14212. This grid is then used to generate a severely ill-conditioned problem (M3Dsm 1) for ∆t = 1 s by changing the values of kclay to 10−11 m/s, and cM to 10−2 MPa−1 . In the second test case, denoted as M3D, a plane triangulation made of 1025 nodes and 2016 triangles projected onto 31 layers is used. The M3D problem totals n = 31775 nodes with N = 127100.

4.1 Numerical Results This section shows some results from both the proposed preconditioners and the ILUT preconditioner applied to the native LSL scaled and reordered system [14] for the test cases described above. All the iterations are completed so as to get a relative error of the same order of magnitude. Test case M3Dsm Table 1 gives the Bi-CGSTAB results using ILUT with optimal parameters after the LSL preliminary scaling. ∆t 0

10 101 102 103 104

nnzprec 879600 879540 879272 1034739 794843

iter 155 163 162 155 150

CPU time (s) preconditioner Bi-CGSTAB 8.08 13.75 7.70 12.04 7.39 12.13 5.75 12.83 3.69 13.13

tot 22.07 19.99 19.77 18.83 17.07

Table 1: Problem M3Dsm. CPU time (s) for Bi-CGSTAB preconditioned with optimal ILUT after the LSL preliminary scaling. The table shows also the number of non zeroes of the preconditioner (nnzprec) and the number of Bi-CGSTAB iterations (iter). Table 2 is the same as Table 1 when the ICP preconditioner is used with G = diag(K). Note the very poor ICP performance related to the bad G−1 quality. The results of Table 3 are obtained with the AINV(0.05) approximation of K −1 and the iterative solution to (12), which is equivalent to using M−1 as the preconditioner for A. As is known from theoretical findings, PCG converges even though the inner system (12) is solved with a low accuracy (10−4 is used as exit tolerance for the relative residual). The number of (outer) iterations is smaller than in Table 1 thus providing evidence of the good spectral properties of the preconditioned matrix. However, the cost required by the inner iterations is quite high, which accounts for the final higher CPU time. 9

∆t

100 101 102 103 104

nnzprec

700211 659975 617941 571540 489685

iter

4049 5654 4481 4583 4909

CPU time (s) preconditioner Bi-CGSTAB preprocessing 1.7 353.9 1.6 482.9 1.5 371.1 1.3 375.0 1.1 394.1

tot 1.35 355.6 484.6 372.7 376.6 395.4

Table 2: Problem M3Dsm. Results for the ICP preconditioner with diagonal approximation of the K and ILUT(10−3 ) approximation of S. ∆t

100 101 102 103 104

nnzprec

1597842 1597825 1597713 1597256 1595899

iter outer (inner) 108 (538) 109 (546) 100 (509) 111 (557) 114 (565)

CPU time (s) preconditioner PCG preprocessing 5.33 40.50 5.61 40.75 5.30 33.54 5.54 43.00 5.61 41.99

tot 17.16 41.13 41.35 34.10 43.60 48.24

Table 3: Problem M3Dsm. Results for the exact constraint preconditioner with AINV(0.05) and the ILU(0)–PCG to solve system (12) (tolcg = 10−4 ). Finally, Tables 4 and 5 present the results of the ICP preconditioner using the ILUT and ILU(0) approximation of S. Of course in this case PCG is not theoretically expected to converge, and again we use Bi-CGSTAB. In either case we obtain an improvement in terms of both number of iterations and CPU time as compared to Table 1. In particular, when ILU(0) is used to approximate S, the CPU time is more than halved, irrespective of the time-step size. Test case M3Dsm 1 For this test case, Bi-CGSTAB, preconditioned with ILUT and matrix A preliminary scaled, could not converge for any choice of the ILUT threshold parameters. By contrast, the ICP preconditioner was successful with the results shown in Table 6. Test case M3D We finally provide in Tables 7 and 8 some preliminary results for the largest test case. Here for small time-steps the new preconditioner gives comparable results to the native Bi-CGSTAB preconditioned with the optimal ILUT, while it appears to be less efficient for large time steps. 10

∆t

100 101 102 103 104

nnzprec iter

1145329 1055471 840659 539813 420948

80 82 81 87 88


tot 17.16 14.74 16.53 14.33 12.04 9.27

Table 4: Problem M3Dsm. Results for the ICP preconditioner with AINV(0.05) and ILUT(10−3 ) approximation of S.

∆t

100 101 102 103 104

nnzprec

666725 602162 464097 376814 337068

iter

84 83 85 87 88


tot 17.16 8.46 8.33 7.41 7.86 7.24

Table 5: Problem M3Dsm. Results for the ICP preconditioner with AINV(0.05) and ILU(0) approximation of S.

11

∆t

100 101 102 103 104

nnzprec

1188099 1140614 1012933 773204 561068

iter

163 184 153 273 224


tot 17.16 32.18 30.77 25.13 35.81 22.41

Table 6: Problem M3Dsm 1. Results for the ICP preconditioner with AINV(0.05) and ILUT(10−3 ) approximation of S. ∆t

100 101 102 103 104

nnzprec

8008091 8007695 8003157 7946785 7680900

iter

234 230 227 239 251


tot 528.40 229.17 231.87 222.03 236.96 246.37

Table 7: Problem M3D. Results for the ICP preconditioner with AINV(0.1) and ILU(0) approximation of S. ∆t 0

10 101 102 103 104

nnzprec 12674810 12635156 12554760 10447813 10614261

iter 178 185 163 150 79

CPU time (s) preconditioner Bi-CGSTAB 71.81 156.90 70.08 169.08 63.57 130.44 40.05 122.43 22.97 67.62

tot 231.21 241.66 196.51 165.98 93.08

Table 8: Problem M3D. Results for the native Bi-CGSTAB preconditioned with the optimal ILUT.

5 Conclusions A novel block ICP (inexact constraint preconditioner) has been developed and efficiently implemented for the iterative solution to FE coupled consolidation equations by using projection conjugate gradient-like methods based on Krylov subspaces. The ICP efficiency depends on the factorized AINV preconditioning for the structural block K and the incomplete Cholesky decomposition for the Schur complement S. The comparison between the Bi-CGSTAB performance obtained by using the ICP 12

and ILUT preconditioner is made on two realistic 3D consolidation problems and shows that the ICP preconditioner allows for the most efficient result in the smaller test case with speed-up ≃ 2, while in the larger one the two approaches exhibit a comparable behavior. The robustness of the ICP preconditioner is also tested in a severely ill-conditioned problem where BiCGSTAB preconditioned with ILUT does not converge. As anticipated from theory, the exact constraint preconditioner allows for convergence of the Conjugate Gradient method for the solution to the indefinite system of discretized consolidation equations. Though theoretically elegant, such a solution proves computationally less efficient than native Bi-CGSTAB. However, the preliminary results discussed above show that the inexact constraint preconditioners are flexible, robust and reliable tools for the iterative solution to FE consolidation models and possess a promising potential for further possible improvements. Acknowledgments. This study has been supported by the Italian MIUR project ”Numerical models for multiphase flow and deformation in porous media”.

References ˚ [1] M. B ENZI , J. K. C ULLUM , AND M. T UMA , Robust approximate inverse preconditioning for the conjugate gradient method, SIAM J. Sci. Comput., 22 (2000), pp. 1318–1332. ˚ [2] M. B ENZI AND M. T UMA , A sparse approximate inverse preconditioner for nonsymmetric linear systems, SIAM J. Sci. Comput., 19 (1998), pp. 968–994. [3]

, A comparative study of sparse approximate inverse preconditioners, Applied Numerical Mathematics, 30 (1999), pp. 305–340.

[4] L. B ERGAMASCHI , J. G ONDZIO , M. V ENTURIN , AND G. Z ILLI, Inexact constraint preconditioners for linear systems arising in interior point methods, Computational Opt. & Appls, (2006). to appear. [5] L. B ERGAMASCHI , J. G ONDZIO , AND G. Z ILLI, Preconditioning indefinite systems in interior point methods for optimization, Computational Opt. & Appls, 28 (2004), pp. 149–171. [6] M. A. B IOT, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), pp. 155–164. [7] J. R. B OOKER AND J. C. S MALL, An investigation of the stability of numerical solutions of Biot’s equations of consolidation, Int. J. Solids Struct., 11 (1975), pp. 907–917. [8] J. C ASTRO, A specialized interior-point algorithm for multicommodity network flows, SIAM Journal on Optimization, 10 (2000), pp. 852–877. 13

[9] S. C HAN , K. P HOON , AND F. L EE, A modified Jacobi preconditioner for solving ill-conditioned Biot’s consolidation equations using symmetric quasi-minimal residual method., Int. J. Numer. Anal. Methods Geomech., 25 (2001), pp. 1001– 1025. [10] X. C HEN , K. C. T OH , AND K. K. P HOON, A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations, International Journal for Numerical Methods in Engineering, 65 (2006), pp. 785–807. [11] M. F ERRONATO , G. G AMBOLATI , AND P. T EATINI, Ill-conditioning of finite element poroelasticity equations., Int. J. Solids Struct., 38 (2001), pp. 5995– 6014. [12] G. G AMBOLATI , G. P INI , AND M. F ERRONATO, Numerical performance of projection methods in finite element consolidation models., Int. J. Numer. Anal. Methods Geomech., 25 (2001), pp. 1429–1447. [13]

, Direct, partitioned and projected solution to finite element consolidation models., Int. J. Numer. Anal. Methods Geomech., 26 (2002), pp. 1371–1383.

[14]

, Scaling improves stability of preconditioned CG-like solvers for FE consolidation equations., Int. J. Numer. Anal. Methods Geomech., 27 (2003), pp. 1043–1056.

[15] G. G AMBOLATI , G. P INI , AND T. T UCCIARELLI, A 3-D finite element conjugate gradient model of subsurface flow with automatic mesh generation., Adv. Water Resources, 9 (1986), pp. 34–41. [16] J. G EERTSMA, Problems of rock mechanics in petroleum production engineering, in Proc. 1st Cong. Int. Soc. Rock Mechan., Lisbon, 1966, pp. 585–594. [17] A. G REENBAUM, Iterative methods for solving linear systems, vol. 17 of Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. [18] M. J. G ROTE AND T. H UCKLE, Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. Comput., 18 (1997), pp. 838–853. [19] L. Y. KOLOTILINA AND A. Y. Y EREMIN, Factorized sparse approximate inverse preconditionings I. Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45–58. [20] M. ROZLO Zˇ N´I K AND V. S IMONCINI, Krylov subspace methods for saddle point problems with indefinite preconditioning, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 368–391 (electronic). [21] K. C. T OH , K. K. P HOON , AND S. H. C HAN, Block preconditioners for symmetric indefinite linear systems, International Journal for Numerical Methods in Engineering, 60 (2004), pp. 1361–1381. 14

[22] W. VAN DER K NAAP, Nonlinear behavior of elastic porous media, Petroleum Transactions of the AIME, 216 (1959), pp. 179–187. [23] H. A. VAN DER VORST, Bi-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 631–644.

15