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August  2021, 29(3): 2517-2532. doi: 10.3934/era.2020127

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

* Corresponding author: Junping Wang

Received  August 2020 Published  August 2021 Early access  December 2020

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.

Citation: Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, 2021, 29 (3) : 2517-2532. doi: 10.3934/era.2020127
References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2 Edition, Academic Press, New York, 2003.   Google Scholar
[2]

I. Babuška, The finite element method with penalty, Math. Comp., 27 (1973), 221-228.  doi: 10.1090/S0025-5718-1973-0351118-5.  Google Scholar

[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.  Google Scholar

[4]

M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), 155-164.  doi: 10.1063/1.1712886.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. Brezzi, On the existence, uniqueness, and approximation of saddle point problems arising from Lagrange multipliers, RAIRO, 8 (1974), 129-151.   Google Scholar

[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.  Google Scholar

[9]

X. HuC. RodrigoF. 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.  Google Scholar

[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.  Google Scholar

[11]

S. KumarR. Oyarz$\acute{u}$aR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

R. LeidermanP. BarboneA. 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.  Google Scholar

[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.  Google Scholar

[18]

M. A. MuradV. 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.  Google Scholar

[19]

J.-C. Nédélec, Mixed finite elements in ${\bf R}^{3}$, Numer. Math., 35 (1980), 315-341.  doi: 10.1007/BF01396415.  Google Scholar

[20]

P. A. NettiL. T. BaxterY. BoucherR. 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.  Google Scholar

[21]

J. A. Nitsche, On Korn's second inequality, RAIRO Anal. Numér., 15 (1981), 237-248.  doi: 10.1051/m2an/1981150302371.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2 Edition, Academic Press, New York, 2003.   Google Scholar
[2]

I. Babuška, The finite element method with penalty, Math. Comp., 27 (1973), 221-228.  doi: 10.1090/S0025-5718-1973-0351118-5.  Google Scholar

[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.  Google Scholar

[4]

M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), 155-164.  doi: 10.1063/1.1712886.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. Brezzi, On the existence, uniqueness, and approximation of saddle point problems arising from Lagrange multipliers, RAIRO, 8 (1974), 129-151.   Google Scholar

[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.  Google Scholar

[9]

X. HuC. RodrigoF. 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.  Google Scholar

[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.  Google Scholar

[11]

S. KumarR. Oyarz$\acute{u}$aR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

R. LeidermanP. BarboneA. 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.  Google Scholar

[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.  Google Scholar

[18]

M. A. MuradV. 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.  Google Scholar

[19]

J.-C. Nédélec, Mixed finite elements in ${\bf R}^{3}$, Numer. Math., 35 (1980), 315-341.  doi: 10.1007/BF01396415.  Google Scholar

[20]

P. A. NettiL. T. BaxterY. BoucherR. 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.  Google Scholar

[21]

J. A. Nitsche, On Korn's second inequality, RAIRO Anal. Numér., 15 (1981), 237-248.  doi: 10.1051/m2an/1981150302371.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Table 1.  Convergence at $ t^n = 1 $ when $ \tau = h $ for $ ([P_2]^2,P_0,P_0, RT_0) $: Example 1 with homogeneous Dirichlet boundary condition
h $ |||e_{\bf{u}}^n||| $ $ ||e_{\bf{u}}^n|| $ $ ||e_z^n|| $ $ ||e_p^n|| $ $ ||e_{{\bf{q}}}^n|| $
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 $ |||e_{\bf{u}}^n||| $ $ ||e_{\bf{u}}^n|| $ $ ||e_z^n|| $ $ ||e_p^n|| $ $ ||e_{{\bf{q}}}^n|| $
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
Table 2.  Convergence at $ t^n = 1 $ when $ h = 1/512 $ for $ ([P_2]^2,P_0,P_0, RT_0) $: Example 1 with homogeneous Dirichlet boundary condition
$ \tau $ $ |||e_{\bf{u}}^n||| $ $ ||e_{\bf{u}}^n|| $ $ ||e_z^n|| $ $ || e_p^n|| $ $ ||e_{{\bf{q}}}^n|| $
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
$ \tau $ $ |||e_{\bf{u}}^n||| $ $ ||e_{\bf{u}}^n|| $ $ ||e_z^n|| $ $ || e_p^n|| $ $ ||e_{{\bf{q}}}^n|| $
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
Table 3.  Convergence at $ t^n = 1 $ when $ \tau = h $ for $ ([P_2]^2,P_0,P_0, RT_0) $: Example 2 with non-homogeneous Dirichlet boundary data
h $ |||e_{\bf{u}}^n||| $ $ ||e_{\bf{u}}^n|| $ $ ||e_z^n|| $ $ ||e_p^n|| $ $ ||e_{{\bf{q}}}^n|| $
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 $ |||e_{\bf{u}}^n||| $ $ ||e_{\bf{u}}^n|| $ $ ||e_z^n|| $ $ ||e_p^n|| $ $ ||e_{{\bf{q}}}^n|| $
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
Table 4.  Convergence at $ t^n = 1 $ when $ \tau = h $ for $ ([P_2]^2,P_0,P_0, RT_0) $: Example 2 with mixed Dirichlet and Neumann boundary conditions
h $ |||e_{\bf{u}}^n||| $ $ ||e_{\bf{u}}^n|| $ $ ||e_z^n|| $ $ ||e_p^n|| $ $ ||e_{{\bf{q}}}^n|| $
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 $ |||e_{\bf{u}}^n||| $ $ ||e_{\bf{u}}^n|| $ $ ||e_z^n|| $ $ ||e_p^n|| $ $ ||e_{{\bf{q}}}^n|| $
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
Table 5.  Convergence at $ t^n = 1 $ when $ \tau = h $ for $ ([P_2]^2,P_1,P_0, RT_0) $: Example 2 with non-homogeneous Dirichlet boundary data
h $ |||e_{\bf{u}}^n||| $ $ ||e_{\bf{u}}^n|| $ $ ||e_z^n|| $ $ ||e_p^n|| $ $ ||e_{{\bf{q}}}^n|| $
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 $ |||e_{\bf{u}}^n||| $ $ ||e_{\bf{u}}^n|| $ $ ||e_z^n|| $ $ ||e_p^n|| $ $ ||e_{{\bf{q}}}^n|| $
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
Table 6.  Convergence at $ t^n = 1 $ when $ \tau = h $ for $ ([P_2]^2,P_1,P_0, RT_0) $: Example 2 with mixed Dirichlet and Neumann boundary data
h $ |||e_{\bf{u}}^n||| $ $ ||e_{\bf{u}}^n|| $ $ ||e_z^n|| $ $ ||e_p^n|| $ $ ||e_{{\bf{q}}}^n|| $
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 $ |||e_{\bf{u}}^n||| $ $ ||e_{\bf{u}}^n|| $ $ ||e_z^n|| $ $ ||e_p^n|| $ $ ||e_{{\bf{q}}}^n|| $
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
[1]

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