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A four-field mixed finite element method for Biot's consolidation problems
1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China |
2. | Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA |
3. | Division of Mathematical Sciences, National Science Foundation, Alexandria, VA 22314, USA |
This article presents a four-field mixed finite element method for Biot's consolidation problems, where the four fields include the displacement, total stress, flux and pressure for the porous medium component of the modeling system. The mixed finite element method involving Raviart-Thomas element is used for the fluid flow equation, while the Crank-Nicolson scheme is employed for the time discretization. The main contribution of this work is the derivation of the optimal order error estimates for semi-discrete and fully-discrete schemes for the unknowns in energy norm or $ L^2 $ norm. Numerical experiments are presented to validate the theoretical results.
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2 Edition, Academic Press, New York, 2003.
![]() |
[2] |
I. Babuška,
The finite element method with penalty, Math. Comp., 27 (1973), 221-228.
doi: 10.1090/S0025-5718-1973-0351118-5. |
[3] |
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), A2222–A2245.
doi: 10.1137/15M1009822. |
[4] |
M. A. Biot,
General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), 155-164.
doi: 10.1063/1.1712886. |
[5] |
M. A. Biot,
Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 26 (1955), 182-185.
doi: 10.1063/1.1721956. |
[6] |
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008.
doi: 10.1007/978-0-387-75934-0. |
[7] |
F. Brezzi,
On the existence, uniqueness, and approximation of saddle point problems arising from Lagrange multipliers, RAIRO, 8 (1974), 129-151.
|
[8] |
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, 15. Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-3172-1. |
[9] |
X. Hu, C. Rodrigo, F. J. Gaspar and L. T. Zikatanov,
A nonconforming finite element method for the Biot's consolidation model in poroelasticity, J. Comput. Appl. Math., 310 (2017), 143-154.
doi: 10.1016/j.cam.2016.06.003. |
[10] |
J. Korsawe and G. Starke,
A least-squares mixed finite element method for Biot's consolidation problem in porous media, SIAM J. Numer. Anal., 43 (2005), 318-339.
doi: 10.1137/S0036142903432929. |
[11] |
S. Kumar, R. Oyarz$\acute{u}$a, R. Ruiz-Baier and R. Sandilya,
Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity, ESAIM Math. Model. Numer. Anal., 54 (2020), 273-299.
doi: 10.1051/m2an/2019063. |
[12] |
J. J. Lee,
Robust error analysis of coupled mixed methods for Biot's consolidation model, J. Sci. Comput., 69 (2016), 610-632.
doi: 10.1007/s10915-016-0210-0. |
[13] |
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), A1–A24.
doi: 10.1137/15M1029473. |
[14] |
J. J. Lee, E. Piersanti, K.-A. Mardal and M. E. Rognes, A mixed finite element method for nearly incompressible multiple-network poroelasticity, SIAM J. Sci. Comput., 41 (2019), A722–A747.
doi: 10.1137/18M1182395. |
[15] |
R. Leiderman, P. Barbone, A. Oberai and J. Bamber, Coupling between elastic strain and interstitial fluid flow: ramifications for poroelastic imaging, Phys. Med. Biol., 51 (2006), 6291-6313. Google Scholar |
[16] |
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), 359-382.
doi: 10.1016/0045-7825(92)90193-N. |
[17] |
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), 645-667.
doi: 10.1002/nme.1620370407. |
[18] |
M. A. Murad, V. Thom$\acute{e}$e and A. F. D. Loula,
Asymptotic behavior of semidiscrete finite-element approximations of Biot's consolidation problem, SIAM J. Numer. Anal., 33 (1996), 1065-1083.
doi: 10.1137/0733052. |
[19] |
J.-C. Nédélec,
Mixed finite elements in ${\bf R}^{3}$, Numer. Math., 35 (1980), 315-341.
doi: 10.1007/BF01396415. |
[20] |
P. A. Netti, L. T. Baxter, Y. Boucher, R. Skalak and R. K. Jain,
Macro-and microscopic fluid transport in living tissues: Application to solid tumors, AIChE Journal of Bioengineering Food, and Natural Products, 43 (1997), 818-834.
doi: 10.1002/aic.690430327. |
[21] |
J. A. Nitsche,
On Korn's second inequality, RAIRO Anal. Numér., 15 (1981), 237-248.
doi: 10.1051/m2an/1981150302371. |
[22] |
R. Oyarz$\acute{u}$a and R. Ruiz-Baier,
Locking-free finite element methods for poroelasticity, SIAM J. Numer. Anal., 54 (2016), 2951-2973.
doi: 10.1137/15M1050082. |
[23] |
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), 131-144.
doi: 10.1007/s10596-007-9045-y. |
[24] |
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), 145-158.
doi: 10.1007/s10596-007-9044-z. |
[25] |
W. Qi, P. Seshaiyer and J. Wang, Finite element method with the total stress variable for Biot's consolidation model, 2020, https://arXiv.org/abs/2008.01278. Google Scholar |
[26] |
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin, 1994. |
[27] |
P.-A. Raviart and J. M. Thomas, A mixed finite element method for second order elliptic problems, Mathematical Aspects of the Finite Element Method (1. Galligani, E. Magenes, eds.), Lectures Notes in Math., Springer-Verlag, New York, 606 (1977), 292–315. |
[28] |
S.-Y. Yi,
Convergence analysis of a new mixed finite element method for Biot's consolidation model, Numer. Methods Partial Differential Equations, 30 (2014), 1189-1210.
doi: 10.1002/num.21865. |
show all references
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2 Edition, Academic Press, New York, 2003.
![]() |
[2] |
I. Babuška,
The finite element method with penalty, Math. Comp., 27 (1973), 221-228.
doi: 10.1090/S0025-5718-1973-0351118-5. |
[3] |
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), A2222–A2245.
doi: 10.1137/15M1009822. |
[4] |
M. A. Biot,
General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), 155-164.
doi: 10.1063/1.1712886. |
[5] |
M. A. Biot,
Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 26 (1955), 182-185.
doi: 10.1063/1.1721956. |
[6] |
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008.
doi: 10.1007/978-0-387-75934-0. |
[7] |
F. Brezzi,
On the existence, uniqueness, and approximation of saddle point problems arising from Lagrange multipliers, RAIRO, 8 (1974), 129-151.
|
[8] |
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, 15. Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-3172-1. |
[9] |
X. Hu, C. Rodrigo, F. J. Gaspar and L. T. Zikatanov,
A nonconforming finite element method for the Biot's consolidation model in poroelasticity, J. Comput. Appl. Math., 310 (2017), 143-154.
doi: 10.1016/j.cam.2016.06.003. |
[10] |
J. Korsawe and G. Starke,
A least-squares mixed finite element method for Biot's consolidation problem in porous media, SIAM J. Numer. Anal., 43 (2005), 318-339.
doi: 10.1137/S0036142903432929. |
[11] |
S. Kumar, R. Oyarz$\acute{u}$a, R. Ruiz-Baier and R. Sandilya,
Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity, ESAIM Math. Model. Numer. Anal., 54 (2020), 273-299.
doi: 10.1051/m2an/2019063. |
[12] |
J. J. Lee,
Robust error analysis of coupled mixed methods for Biot's consolidation model, J. Sci. Comput., 69 (2016), 610-632.
doi: 10.1007/s10915-016-0210-0. |
[13] |
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), A1–A24.
doi: 10.1137/15M1029473. |
[14] |
J. J. Lee, E. Piersanti, K.-A. Mardal and M. E. Rognes, A mixed finite element method for nearly incompressible multiple-network poroelasticity, SIAM J. Sci. Comput., 41 (2019), A722–A747.
doi: 10.1137/18M1182395. |
[15] |
R. Leiderman, P. Barbone, A. Oberai and J. Bamber, Coupling between elastic strain and interstitial fluid flow: ramifications for poroelastic imaging, Phys. Med. Biol., 51 (2006), 6291-6313. Google Scholar |
[16] |
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), 359-382.
doi: 10.1016/0045-7825(92)90193-N. |
[17] |
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), 645-667.
doi: 10.1002/nme.1620370407. |
[18] |
M. A. Murad, V. Thom$\acute{e}$e and A. F. D. Loula,
Asymptotic behavior of semidiscrete finite-element approximations of Biot's consolidation problem, SIAM J. Numer. Anal., 33 (1996), 1065-1083.
doi: 10.1137/0733052. |
[19] |
J.-C. Nédélec,
Mixed finite elements in ${\bf R}^{3}$, Numer. Math., 35 (1980), 315-341.
doi: 10.1007/BF01396415. |
[20] |
P. A. Netti, L. T. Baxter, Y. Boucher, R. Skalak and R. K. Jain,
Macro-and microscopic fluid transport in living tissues: Application to solid tumors, AIChE Journal of Bioengineering Food, and Natural Products, 43 (1997), 818-834.
doi: 10.1002/aic.690430327. |
[21] |
J. A. Nitsche,
On Korn's second inequality, RAIRO Anal. Numér., 15 (1981), 237-248.
doi: 10.1051/m2an/1981150302371. |
[22] |
R. Oyarz$\acute{u}$a and R. Ruiz-Baier,
Locking-free finite element methods for poroelasticity, SIAM J. Numer. Anal., 54 (2016), 2951-2973.
doi: 10.1137/15M1050082. |
[23] |
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), 131-144.
doi: 10.1007/s10596-007-9045-y. |
[24] |
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), 145-158.
doi: 10.1007/s10596-007-9044-z. |
[25] |
W. Qi, P. Seshaiyer and J. Wang, Finite element method with the total stress variable for Biot's consolidation model, 2020, https://arXiv.org/abs/2008.01278. Google Scholar |
[26] |
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin, 1994. |
[27] |
P.-A. Raviart and J. M. Thomas, A mixed finite element method for second order elliptic problems, Mathematical Aspects of the Finite Element Method (1. Galligani, E. Magenes, eds.), Lectures Notes in Math., Springer-Verlag, New York, 606 (1977), 292–315. |
[28] |
S.-Y. Yi,
Convergence analysis of a new mixed finite element method for Biot's consolidation model, Numer. Methods Partial Differential Equations, 30 (2014), 1189-1210.
doi: 10.1002/num.21865. |
h | ||||||||||
error | order | error | order | error | order | error | order | error | order | |
1/8 | 1.0970e+00 | 3.6212e-02 | 9.8472e-04 | 1.0326e-03 | 2.6009e-04 | |||||
1/16 | 5.5495e-01 | 0.98 | 9.1920e-03 | 1.98 | 2.9921e-04 | 1.72 | 3.4406e-04 | 1.59 | 1.1141e-04 | 1.22 |
1/32 | 2.7839e-01 | 1.00 | 2.3020e-03 | 2.00 | 1.0632e-04 | 1.49 | 1.1722e-04 | 1.55 | 3.6399e-05 | 1.61 |
1/64 | 1.3934e-01 | 1.00 | 5.7569e-04 | 2.00 | 2.8200e-05 | 1.92 | 3.0540e-05 | 1.94 | 8.6783e-06 | 2.02 |
h | ||||||||||
error | order | error | order | error | order | error | order | error | order | |
1/8 | 1.0970e+00 | 3.6212e-02 | 9.8472e-04 | 1.0326e-03 | 2.6009e-04 | |||||
1/16 | 5.5495e-01 | 0.98 | 9.1920e-03 | 1.98 | 2.9921e-04 | 1.72 | 3.4406e-04 | 1.59 | 1.1141e-04 | 1.22 |
1/32 | 2.7839e-01 | 1.00 | 2.3020e-03 | 2.00 | 1.0632e-04 | 1.49 | 1.1722e-04 | 1.55 | 3.6399e-05 | 1.61 |
1/64 | 1.3934e-01 | 1.00 | 5.7569e-04 | 2.00 | 2.8200e-05 | 1.92 | 3.0540e-05 | 1.94 | 8.6783e-06 | 2.02 |
error | order | error | order | error | order | error | order | |||
1 | 1.9294e+00 | 3.1761e-01 | 5.6108e-02 | 6.9489e-02 | 1.4643e-02 | |||||
1/2 | 4.7787e-01 | 2.01 | 7.8199e-02 | 2.02 | 1.3792e-02 | 2.02 | 1.7129e-02 | 2.02 | 3.3258e-03 | 2.14 |
1/4 | 1.2053e-01 | 1.99 | 1.9539e-02 | 2.00 | 3.4454e-03 | 2.00 | 4.2784e-03 | 2.00 | 8.3143e-04 | 2.00 |
1/8 | 3.4530e-02 | 1.80 | 4.8821e-03 | 2.00 | 8.6090e-04 | 2.00 | 1.0691e-03 | 2.00 | 2.0775e-04 | 2.00 |
error | order | error | order | error | order | error | order | |||
1 | 1.9294e+00 | 3.1761e-01 | 5.6108e-02 | 6.9489e-02 | 1.4643e-02 | |||||
1/2 | 4.7787e-01 | 2.01 | 7.8199e-02 | 2.02 | 1.3792e-02 | 2.02 | 1.7129e-02 | 2.02 | 3.3258e-03 | 2.14 |
1/4 | 1.2053e-01 | 1.99 | 1.9539e-02 | 2.00 | 3.4454e-03 | 2.00 | 4.2784e-03 | 2.00 | 8.3143e-04 | 2.00 |
1/8 | 3.4530e-02 | 1.80 | 4.8821e-03 | 2.00 | 8.6090e-04 | 2.00 | 1.0691e-03 | 2.00 | 2.0775e-04 | 2.00 |
h | ||||||||||
error | order | error | order | error | order | error | order | error | order | |
1/8 | 1.6784e-02 | 5.6421e-04 | 1.9561e-02 | 1.2901e-03 | 6.7991e-03 | |||||
1/16 | 8.6688e-03 | 0.95 | 1.4538e-04 | 1.96 | 5.3817e-03 | 1.86 | 4.4940e-04 | 1.52 | 2.3988e-03 | 1.50 |
1/32 | 4.3990e-03 | 0.98 | 3.6669e-05 | 1.99 | 1.4495e-03 | 1.89 | 9.9263e-05 | 2.18 | 7.0921e-04 | 1.76 |
1/64 | 2.2148e-03 | 0.99 | 9.1937e-06 | 2.00 | 3.9949e-04 | 1.86 | 2.2901e-05 | 2.12 | 2.2685e-04 | 1.64 |
h | ||||||||||
error | order | error | order | error | order | error | order | error | order | |
1/8 | 1.6784e-02 | 5.6421e-04 | 1.9561e-02 | 1.2901e-03 | 6.7991e-03 | |||||
1/16 | 8.6688e-03 | 0.95 | 1.4538e-04 | 1.96 | 5.3817e-03 | 1.86 | 4.4940e-04 | 1.52 | 2.3988e-03 | 1.50 |
1/32 | 4.3990e-03 | 0.98 | 3.6669e-05 | 1.99 | 1.4495e-03 | 1.89 | 9.9263e-05 | 2.18 | 7.0921e-04 | 1.76 |
1/64 | 2.2148e-03 | 0.99 | 9.1937e-06 | 2.00 | 3.9949e-04 | 1.86 | 2.2901e-05 | 2.12 | 2.2685e-04 | 1.64 |
h | ||||||||||
error | order | error | order | error | order | error | order | error | order | |
1/8 | 1.6863e-02 | 5.3041e-04 | 2.0902e-02 | 1.1893e-03 | 6.4497e-03 | |||||
1/16 | 8.6834e-03 | 0.96 | 1.3557e-04 | 1.97 | 5.5918e-03 | 1.90 | 2.8353e-04 | 2.07 | 1.9159e-03 | 1.75 |
1/32 | 4.4019e-03 | 0.98 | 3.4070e-05 | 1.99 | 1.4745e-03 | 1.92 | 3.4283e-05 | 3.05 | 4.1519e-04 | 2.21 |
1/64 | 2.2155e-03 | 0.99 | 8.5408e-06 | 2.00 | 4.0097e-04 | 1.88 | 5.5825e-06 | 2.62 | 9.2273e-05 | 2.17 |
h | ||||||||||
error | order | error | order | error | order | error | order | error | order | |
1/8 | 1.6863e-02 | 5.3041e-04 | 2.0902e-02 | 1.1893e-03 | 6.4497e-03 | |||||
1/16 | 8.6834e-03 | 0.96 | 1.3557e-04 | 1.97 | 5.5918e-03 | 1.90 | 2.8353e-04 | 2.07 | 1.9159e-03 | 1.75 |
1/32 | 4.4019e-03 | 0.98 | 3.4070e-05 | 1.99 | 1.4745e-03 | 1.92 | 3.4283e-05 | 3.05 | 4.1519e-04 | 2.21 |
1/64 | 2.2155e-03 | 0.99 | 8.5408e-06 | 2.00 | 4.0097e-04 | 1.88 | 5.5825e-06 | 2.62 | 9.2273e-05 | 2.17 |
h | ||||||||||
error | order | error | order | error | order | error | order | error | order | |
1/8 | 3.1646e-04 | 1.3523e-05 | 5.1888e-02 | 3.2782e-03 | 1.6614e-02 | |||||
1/16 | 5.1165e-05 | 2.63 | 4.1755e-06 | 1.70 | 1.3362e-02 | 1.96 | 1.1749e-03 | 1.48 | 6.1427e-03 | 1.44 |
1/32 | 1.0911e-05 | 2.23 | 7.0520e-07 | 2.57 | 3.3167e-03 | 2.01 | 2.5536e-04 | 2.20 | 1.8480e-03 | 1.73 |
1/64 | 3.3563e-06 | 1.70 | 1.2494e-07 | 2.50 | 8.2329e-04 | 2.01 | 5.7363e-05 | 2.15 | 6.1243e-04 | 1.59 |
h | ||||||||||
error | order | error | order | error | order | error | order | error | order | |
1/8 | 3.1646e-04 | 1.3523e-05 | 5.1888e-02 | 3.2782e-03 | 1.6614e-02 | |||||
1/16 | 5.1165e-05 | 2.63 | 4.1755e-06 | 1.70 | 1.3362e-02 | 1.96 | 1.1749e-03 | 1.48 | 6.1427e-03 | 1.44 |
1/32 | 1.0911e-05 | 2.23 | 7.0520e-07 | 2.57 | 3.3167e-03 | 2.01 | 2.5536e-04 | 2.20 | 1.8480e-03 | 1.73 |
1/64 | 3.3563e-06 | 1.70 | 1.2494e-07 | 2.50 | 8.2329e-04 | 2.01 | 5.7363e-05 | 2.15 | 6.1243e-04 | 1.59 |
h | ||||||||||
error | order | error | order | error | order | error | order | error | order | |
1/8 | 5.4994e-04 | 2.8159e-05 | 5.1568e-02 | 3.1130e-03 | 1.4807e-02 | |||||
1/16 | 9.7846e-05 | 2.49 | 6.4894e-06 | 2.12 | 1.3027e-02 | 1.98 | 7.7273e-04 | 2.01 | 4.4788e-03 | 1.73 |
1/32 | 1.8542e-05 | 2.40 | 8.1614e-07 | 2.99 | 3.1901e-03. | 2.03 | 9.6270e-05 | 3.00 | 8.9236e-04 | 2.33 |
1/64 | 4.2955e-06 | 2.11 | 1.0693e-07 | 2.93 | 7.8825e-04. | 2.02 | 1.1182e-05 | 3.11 | 1.7505e-04 | 2.35 |
h | ||||||||||
error | order | error | order | error | order | error | order | error | order | |
1/8 | 5.4994e-04 | 2.8159e-05 | 5.1568e-02 | 3.1130e-03 | 1.4807e-02 | |||||
1/16 | 9.7846e-05 | 2.49 | 6.4894e-06 | 2.12 | 1.3027e-02 | 1.98 | 7.7273e-04 | 2.01 | 4.4788e-03 | 1.73 |
1/32 | 1.8542e-05 | 2.40 | 8.1614e-07 | 2.99 | 3.1901e-03. | 2.03 | 9.6270e-05 | 3.00 | 8.9236e-04 | 2.33 |
1/64 | 4.2955e-06 | 2.11 | 1.0693e-07 | 2.93 | 7.8825e-04. | 2.02 | 1.1182e-05 | 3.11 | 1.7505e-04 | 2.35 |
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