# Abstract

We discuss the construction of robust preconditioners for finite element approximations of Biot’s consolidation model in poroelasticity. More precisely, we study finite element methods based on generalizations of the Hellinger–Reissner principle of linear elasticity, where the stress tensor is one of the unknowns. The Biot model has a number of applications in science, medicine, and engineering. A challenge in many of these applications is that the model parameters range over several orders of magnitude. Therefore, discretization procedures which are well behaved with respect to such variations are needed. The focus of the present paper will be on the construction of preconditioners, such that the preconditioned discrete systems are well-conditioned with respect to variations of the model parameters as well as refinements of the discretization. As a byproduct, we also obtain preconditioners for linear elasticity that are robust in the incompressible limit.

MSC 2010: 65N30; 65N55; 74S05

Funding statement: The research leading to these results has received funding the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 339643. The work of Kent-Andre Mardal has also been supported by the Research Council of Norway through grant no. 179578/F50.

## A Appendix: Proofs of (3.3) and (3.5)

Here we provide proofs of inequalities (3.3) and (3.5).

## Proof of (3.3).

Fix τΣ and recall that |Γt|>0 and τν^=0 on Γt. By the pointwise decomposition

τ=PDτ+(I-PD)τ,

and the fact that (I-PD)τ=1ntrτ𝕀, it suffices to show that

trτ02C((12μPDτ,PDτ)+𝐝𝐢𝐯τ02)

for some constant C independent of τ. To prove this, we use a well-known result for the right inverse of the divergence operator: There exists ϕHΓd1(Ω;𝕍):={φH1(Ω;𝕍):φ|Γd=0} such that

(A.1)

divϕ=trτ,ϕ1Ctrτ0

with C>0 independent of τ, cf. Appendix B. We then have that

trτ02=(trτ,divϕ)=(trτ𝕀,𝐠𝐫𝐚𝐝ϕ).

Since trτ𝕀=n(τ-PDτ), we get

trτ02=n(τ,𝐠𝐫𝐚𝐝ϕ)-n(PDτ,𝐠𝐫𝐚𝐝ϕ)
=-n(𝐝𝐢𝐯τ,ϕ)-n(PDτ,𝐠𝐫𝐚𝐝ϕ),

where the first term of the final form is a result of integration by parts. Next, we may use the Cauchy–Schwarz inequality, which results in

n(𝐝𝐢𝐯τ02+PDτ02)12ϕ1
C(𝐝𝐢𝐯τ02+PDτ02)12trτ0,

and so the result follows after dividing by trτ0. ∎

When |Γt|=0, i.e., Γd=Ω, (A.1) can only hold if trτ has mean value zero. However, with this constraint, we can prove (3.5) with almost the same argument as above.

## Proof of (3.5).

Fix any τΣ. From the decomposition P0τ=PDτ+(P0-PD)τ, it suffices to prove the estimate for (P0-PD)τ component. Denoting the mean value of the trace by

trτ¯:=1|Ω|Ωtrτdx,

we have that (I-PD)P0τ=(P0-PD)τ=1n(trτ-trτ¯)𝕀, and so it is sufficient to show that

trτ-trτ¯02C((12μPDτ,PDτ)+𝐝𝐢𝐯τ02).

Since trτ-trτ¯ is mean-value zero, there exists ϕ{φH1(Ω;𝕍):φ|Ω=0} such that

divϕ=trτ-trτ¯,ϕ1Ctrτ-trτ¯0

with C>0 independent of τ (cf. [14, Theorem 5.1]). The rest of the proof is completely analogous to the proof of (3.3) above. ∎

## B Appendix: Right Inverse of Divergence Operator

A result for the right inverse of the divergence operator, as expressed by (A.1), is closely related to the inf-sup condition for the Stokes problem, and therefore well-known. However, we are not aware of a proper reference for the case when |Ω|>|Γt|>0, i.e., for the case when |Γd|>0, but Γd is not all of Ω. Therefore, for completeness, we include a proof here.

## Lemma B.1.

Assume that |Γt|>0 and set HΓd1(Ω;V)={ϕH1(Ω;V):ϕ|Γd=0}. Then there is a constant C>0 so that for every fL2(Ω) there is a ϕHΓd1(Ω;V) so that

divϕ=f,ϕ1Cf0.

## Proof.

Take any fL2(Ω). We first decompose f into its mean value zero- and mean value part as f=f0+fc, where f0L02(Ω) and fc=af1Ω for af. Further, we can decompose HΓd1(Ω;𝕍)=H01(Ω;𝕍)V1, where

V1:={ϕHΓd1(Ω;𝕍):(𝐠𝐫𝐚𝐝ϕ,𝐠𝐫𝐚𝐝ψ)=0 for all ψH01(Ω;𝕍)}.

Consider then the problem of finding ζV1 so that

(B.1)(𝐠𝐫𝐚𝐝ζ,𝐠𝐫𝐚𝐝ψ)=(𝕀,𝐠𝐫𝐚𝐝ψ)for all ψV1.

By the Lax–Milgram lemma (cf. e.g. [8, Theorem 4.1.6]), problem (B.1) has a unique solution ζ and ζ1C1 for some constant C1>0 depending on Ω. Taking ψ=ζ in (B.1), we obtain

Ωdivζdx=𝐠𝐫𝐚𝐝ζ02.

Therefore, if we set

ω=af𝐠𝐫𝐚𝐝ζ02ζ,

we have

Ωdivωdx=af

and ω1Cfc0 for some constant C depending on ζ. It follows that f-divωL02(Ω), i.e., f-divω has mean value zero. From the theory of Stokes equation, we can thus find a ω0H01(Ω;𝕍) so that

(B.2)divω0=f-divω,ω01C2f-divω0,

where the constant C2 is independent of f-divω (cf. [14, Theorem 5.1]). We set ϕ=ω0+ω, and it follows from (B.2) that divϕ=f. Using the triangle inequality, (B.2) and the properties of ω, we estimate ϕ1 as

ϕ1ω01+ω1C(f-divω0+fc0)C(f0+ω1)Cf0,

which completes the proof. ∎

### References

 D. N. Arnold, F. Brezzi, J. Douglas, Jr., PEERS: A new mixed finite element for plane elasticity, Japan J. Appl. Math. 1 (1984), no. 2, 347–367. 10.1007/BF03167064Search in Google Scholar

 D. N. Arnold, J. Douglas, Jr. and C. P. Gupta, A family of higher order mixed finite element methods for plane elasticity, Numer. Math. 45 (1984), no. 1, 1–22. 10.1007/BF01379659Search in Google Scholar

 D. N. Arnold, R. S. Falk and R. Winther, Preconditioning in H(div) and applications, Math. Comp. 66 (1997), no. 219, 957–984. 10.1090/S0025-5718-97-00826-0Search in Google Scholar

 D. N. Arnold, R. S. Falk and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp. 76 (2007), no. 260, 1699–1723. 10.1090/S0025-5718-07-01998-9Search in Google Scholar

 O. Axelsson, R. Blaheta and P. Byczanski, Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices, Comput. Vis. Sci. 15 (2012), no. 4, 191–207. 10.1007/s00791-013-0209-0Search in Google Scholar

 L. Berger, R. Bordas, D. Kay and S. Tavener, Stabilized lowest-order finite element approximation for linear three-field poroelasticity, SIAM J. Sci. Comput. 37 (2015), no. 5, A2222–A2245. 10.1137/15M1009822Search in Google Scholar

 D. Boffi, F. Brezzi and M. Fortin, Reduced symmetry elements in linear elasticity, Commun. Pure Appl. Anal. 8 (2009), no. 1, 95–121. 10.3934/cpaa.2009.8.95Search in Google Scholar

 D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer Ser. Comput. Math. 44, Springer, Heidelberg, 2013. 10.1007/978-3-642-36519-5Search in Google Scholar

 J. H. Bramble, Multigrid Methods, Pitman Res. Notes in Math. 294, Longman Scientific & Technical, Harlow, 1993. Search in Google Scholar

 Y. Chen, Y. Luo and M. Feng, Analysis of a discontinuous Galerkin method for the Biot’s consolidation problem, Appl. Math. Comput. 219 (2013), no. 17, 9043–9056. 10.1016/j.amc.2013.03.104Search in Google Scholar

 B. Cockburn, J. Gopalakrishnan and J. Guzmán, A new elasticity element made for enforcing weak stress symmetry, Math. Comp. 79 (2010), no. 271, 1331–1349. 10.1090/S0025-5718-10-02343-4Search in Google Scholar

 O. Coussy, Poromechanics, John Wiley & Sons, Hoboken, 2004. 10.1002/0470092718Search in Google Scholar

 M. Farhloul and M. Fortin, Dual hybrid methods for the elasticity and the Stokes problems: A unified approach, Numer. Math. 76 (1997), no. 4, 419–440. 10.1007/s002110050270Search in Google Scholar

 V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Springer Ser. Comput. Math. 5, Springer, Berlin, 1986. 10.1007/978-3-642-61623-5Search in Google Scholar

 J. Gopalakrishnan and J. Guzmán, A second elasticity element using the matrix bubble, IMA J. Numer. Anal. 32 (2012), no. 1, 352–372. 10.1093/imanum/drq047Search in Google Scholar

 J. B. Haga, H. Osnes and H. P. Langtangen, A parallel block preconditioner for large-scale poroelasticity with highly heterogeneous material parameters, Comput. Geosci. 16 (2012), no. 3, 723–734. 10.1007/s10596-012-9284-4Search in Google Scholar

 R. Hiptmair and J. Xu, Nodal auxiliary space preconditioning in H(curl) and H(div) spaces, SIAM J. Numer. Anal. 45 (2007), no. 6, 2483–2509. 10.1137/060660588Search in Google Scholar

 J. J. Lee, Robust error analysis of coupled mixed methods for Biot’s consolidation model, J. Sci. Comput. 69 (2016), no. 2, 610–632. 10.1007/s10915-016-0210-0Search in Google Scholar

 J. J. Lee, Towards a unified analysis of mixed methods for elasticity with weakly symmetric stress, Adv. Comput. Math. 42 (2016), no. 2, 361–376. 10.1007/s10444-015-9427-ySearch in Google Scholar

 J. J. Lee, K.-A. Mardal and R. Winther, Parameter-robust discretization and preconditioning of Biot’s consolidation model, SIAM J. Sci. Comput. 39 (2017), no. 1, A1–A24. 10.1137/15M1029473Search in Google Scholar

 K.-A. Mardal and R. Winther, Preconditioning discretizations of systems of partial differential equations, Numer. Linear Algebra Appl. 18 (2011), no. 1, 1–40. 10.1002/nla.716Search in Google Scholar

 M. A. Murad and A. F. D. Loula, Improved accuracy in finite element analysis of Biot’s consolidation problem, Comput. Methods Appl. Mech. Engrg. 95 (1992), no. 3, 359–382. 10.1016/0045-7825(92)90193-NSearch in Google Scholar

 M. A. Murad and A. F. D. Loula, On stability and convergence of finite element approximations of Biot’s consolidation problem, Internat. J. Numer. Methods Engrg. 37 (1994), no. 4, 645–667. 10.1002/nme.1620370407Search in Google Scholar

 P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: The continuous in time case, Comput. Geosci. 11 (2007), no. 2, 131–144. 10.1007/s10596-007-9045-ySearch in Google Scholar

 P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: The discrete-in-time case, Comput. Geosci. 11 (2007), no. 2, 145–158. 10.1007/s10596-007-9044-zSearch in Google Scholar

 P. J. Phillips and M. F. Wheeler, A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity, Comput. Geosci. 12 (2008), no. 4, 417–435. 10.1007/s10596-008-9082-1Search in Google Scholar

 M. B. Reed, An investigation of numerical errors in the analysis of consolidation by finite elements, Internat. J. Numer. Analyt. Methods Geomech. 8 (1984), no. 3, 243–257. 10.1002/nag.1610080304Search in Google Scholar

 S. Rhebergen, G. N. Wells, A. J. Wathen and R. F. Katz, Three-field block preconditioners for models of coupled magma/mantle dynamics, SIAM J. Sci. Comput. 37 (2015), no. 5, A2270–A2294. 10.1137/14099718XSearch in Google Scholar

 R. E. Showalter, Diffusion in poro-elastic media, J. Math. Anal. Appl. 251 (2000), no. 1, 310–340. 10.1006/jmaa.2000.7048Search in Google Scholar

 J. H. Smith and J. A. Humphrey, Interstitial transport and transvascular fluid exchange during infusion into brain and tumor tissue, Microvasc. Res. 73 (2007), no. 1, 58–73. 10.1016/j.mvr.2006.07.001Search in Google Scholar PubMed

 R. Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513–538. 10.1007/BF01397550Search in Google Scholar

 K. H. Støverud, M. Alnæs, H. P. Langtangen, V. Haughton and K.-A. Mardal, Poro-elastic modeling of syringomyelia – A systematic study of the effects of pia mater, central canal, median fissure, white and gray matter on pressure wave propagation and fluid movement within the cervical spinal cord, Comput. Methods Biomech. Biomed. Engin. 19 (2016), no. 6, 686–698. 10.1080/10255842.2015.1058927Search in Google Scholar PubMed

 P. A. Vermeer and A. Verruijt, An accuracy condition for consolidation by finite elements, Internat. J. Numer. Analyt. Methods Geomech. 5 (1981), no. 1, 1–14. 10.1002/nag.1610050103Search in Google Scholar

 H. F. Wang, Theory of Linear Poroelasticity, Princeton Ser. Geophys., Princeton University Press, Princeton, 2000. Search in Google Scholar

 S.-Y. Yi, A coupling of nonconforming and mixed finite element methods for Biot’s consolidation model, Numer. Methods Partial Differ. Equ. 29 (2013), no. 5, 1749–1777. 10.1002/num.21775Search in Google Scholar

 S.-Y. Yi, Convergence analysis of a new mixed finite element method for Biot’s consolidation model, Numer. Methods Partial Differ. Equ. 30 (2014), no. 4, 1189–1210. 10.1002/num.21865Search in Google Scholar

 O. C. Zienkiewicz and T. Shiomi, Dynamic behaviour of saturated porous media; the generalized Biot formulation and its numerical solution, Internat. J. Numer. Analyt. Methods Geomech. 8 (1984), no. 1, 71–96. 10.1002/nag.1610080106Search in Google Scholar