Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices. (English) Zbl 1388.74035

Summary: Poroelastic models arise in reservoir modeling and many other important applications. Under certain assumptions, they involve a time-dependent coupled system consisting of Navier-Lamé equations for the displacements, Darcy’s flow equation for the fluid velocity and a divergence constraint equation. Stability for infinite time of the continuous problem and, second and third order accurate, time discretized equations are shown. Methods to handle the lack of regularity at initial times are discussed and illustrated numerically. After discretization, at each time step this leads to a block matrix system in saddle point form. Mixed space discretization methods and a regularization method to stabilize the system and avoid locking in the pressure variable are presented. A certain block matrix preconditioner is shown to cluster the eigenvalues of the preconditioned matrix about the unit value but needs inner iterations for certain matrix blocks. The strong clustering leads to very few outer iterations. Various approaches to construct preconditioners are presented and compared. The sensitivity of the number of outer iterations to the stopping accuracy of the inner iterations is illustrated numerically.


74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74S05 Finite element methods applied to problems in solid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] Biot, MA, General theory of three-dimensional consolidation, J. Appl. Phys., 12, 155-164, (1941) · JFM 67.0837.01
[2] Biot, MA, Theory of elasticity and consolidation for porous anisotropic media, J. Appl. Phys., 26, 182-185, (1955) · Zbl 0067.23603
[3] Ewing, R. (ed.): The Mathematics of Reservoir Simulations, Frontiers in Appl. Math. vol. 1. SIAM, Philadelphia (1984) · Zbl 0338.90047
[4] Discacciati, M; Quarteroni, A, Navier-Stokes/Darcy coupling: modelling, analysis and numerical approximation, Rev. Math. Comput., 22, 315-426, (2009) · Zbl 1172.76050
[5] Lipnikov, K.: Numerical Methods for the Biot Model in Poroelasticity. Ph. D. Thesis, University of Houston (2002)
[6] Kuznetsov, Yu; Prokopenko, A, A new multilevel algebraic preconditioner for the diffusion equation in heterogeneous media, Numer. Linear Algebra Appl., 17, 759-769, (2010) · Zbl 1240.65095
[7] Phillips, P.J.: Finite Element Methods in Linear Poroelasticity: Theoretical and Computational Results. Ph.D. Thesis, The University of Texas at Austin (2005)
[8] Axelsson, O; Blaheta, R, Preconditioning of matrices partitioned in two by two block form: eigenvalue estimates and Schwarz DD for mixed FEM, Numer. Linear Algebra Appl., 17, 787-810, (2010) · Zbl 1240.65090
[9] Axelsson, O, Preconditioners for regularized saddle point matrices, J. Numer. Math., 19, 91-112, (2011) · Zbl 1221.65089
[10] Torquato, S.: Random Heterogeneous Materials. Microstructure and Macroscopic Properties. Springer, New York (2002) · Zbl 0988.74001
[11] Axelsson, O, On iterative solvers in structural mechanics, separate displacement orderings and mixed variable methods, Math. Comput. Simul., 50, 11-30, (1999) · Zbl 1053.74651
[12] Phillips, PJ; Wheeler, MF, Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach, Comput. Geosci., 13, 5-12, (2009) · Zbl 1172.74017
[13] Haga, J.B., Osnes, H., Langtangen, H.P.: On the causes of pressure oscillations in low-permeable porous media. Int. J. Numer. Anal. Methods Geomech. (on line) (2011)
[14] Detournay, E., Cheng, A.H.-D.: Fundamentals of poroelasticity. In: Fairhurst, C. (ed.) Chapter 5 in Comprehensive Rock Engineering Principles, Practice and Projects, Vol. II, Analysis and Design Method. Pergamon Press, pp. 113-171 (1993)
[15] Biot, MA; Willis, DG, The elastic coefficients of the theory of consolidation, J. Appl. Mech., 24, 594-601, (1957)
[16] Terzaghi, K.: Theoretical Soil Mechanics. Wiley, New York (1943)
[17] Axelsson, O; Gololobov, SV, Stability and error estimates for the \( θ \)-method for strongly monotone and infinitely stiff evolution equations, Numer. Math., 89, 31-48, (2001) · Zbl 0994.65094
[18] Showalter, RE, Diffusion in poro-elastic media, J. Math. Anal. Appl., 251, 310-340, (2000) · Zbl 0979.74018
[19] Helfrisch, HP, Fehlerabschätzungen für das galerkinverfahren zur Lösung von evolutionsgleichungen, Manuscripta Math., 13, 219-235, (1974) · Zbl 0323.65037
[20] Rannacher, R, Finite element solution of diffusion problems with irregular data, Numer. Math., 43, 309-327, (1984) · Zbl 0512.65082
[21] Murad, MA; Thomée, V; Loula, AFD, Asymptotic behaviour of semidiscrete finite element approximations of biot’s consolidation problem, SIAM J. Numer. Anal., 33, 1065-1083, (1996) · Zbl 0854.76053
[22] Murad, MA, AFD loula. on stability and convergence of finite element approximations of biot’s consolidation problem, Int. J. Numer. Method Eng., 27, 645-667, (1994) · Zbl 0791.76047
[23] Gear, CW; Petzold, LR, ODE methods for the solution of differential/algebraic systems, SIAM J. Numer. Anal., 23, 837-852, (1986) · Zbl 0635.65084
[24] Butcher, J.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2003) · Zbl 1040.65057
[25] Axelsson, O, A class of \( A-\)stable methods, BIT, 9, 185-199, (1969) · Zbl 0208.41504
[26] Axelsson, O.: On the efficiency of a class of \(A-\)stable methods. BIT 14, 279-287 (1974) · Zbl 0289.65028
[27] Axelsson, O., Blaheta, R., Sysala, S., Ahmad, B.: On the solution of high order stable time integration methods. J. Boundary Value Probl. Springer open 108 (2013). doi:10.1186/1687-2770-2013-108 · JFM 67.0837.01
[28] Babuska, I, Error-bounds for the finite element method, Numer. Math., 16, 322-333, (1971) · Zbl 0214.42001
[29] Brezzi, F, On the existence, uniqueness and approximation of saddle point problems, arising from Lagrangian multipliers, R.A.I.R.O. Anal. Numer., 8, 129-151, (1974) · Zbl 0338.90047
[30] Raviart, RA; Thomas, JM, Mixed finite element method for second order elliptic problems, Math. Aspects Finite Element Method Lect. Notes Math., 606, 292-315, (1977)
[31] Axelsson, O; Barker, VA; Neytcheva, M; Polman, B, Solving the Stokes problem on a massively parallel computer, Math. Model. Anal., 4, 1-22, (2000) · Zbl 1162.76332
[32] Sagae, M; Tanabe, K, Upper and lower bounds for the arithmetic-geometric-harmonic means of positive definite matrices, Linear Multilinear Algebra, 37, 279-282, (1994) · Zbl 0816.15017
[33] Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1994) · Zbl 0795.65014
[34] Haga, JB; Osnes, H; Langtangen, HP, Efficient block preconditioners for the coupled equations of pressure and deformation in highly discontinuous media, Int. J. Numer. Anal. Methods Geomech., 35, 1466-1482, (2011)
[35] Saad, Y, A flexible inner-outer preconditioned GMRES-algorithm, SIAM J. Sci. Comput., 14, 461-469, (1993) · Zbl 0780.65022
[36] Axelsson, O; Blaheta, R; Neytcheva, M, Preconditioning for boundary value problems using elementwise Schur complements, SIAM J. Matrix Anal. Appl., 31, 767-789, (2009) · Zbl 1194.65047
[37] Haga, JB; Langtangen, HP; Osnes, H, A parallel block preconditioner for large scale poroelasticity with highly heterogeneous material parameters, Comput. Geosci., 16, 723-734, (2012)
[38] Arbenz, P., Turan, E.: Preconditioning for large scale micro finite element analyses of 3D poroelasticity. In: Manninen, P., Öster, P. (eds.) Applied Parallel and Scientific Computing (PARA 2012), Lecture Notes in Computer Science 7782, pp. 361-374. Springer, Heidelberg (2013)
[39] Ferronato, M; Castelletto, N; Gambolati, G, A fully coupled 3-D mixed finite element model of Biot consolidation, J. Comput. Phys., 229, 4813-4830, (2010) · Zbl 1305.76055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.