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  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, 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]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095

[2]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120

[3]

Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097

[4]

Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019

[5]

Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126

[6]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089

[7]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351

[8]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[9]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[10]

P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178

[11]

Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096

[12]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[13]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051

[14]

Xiaoli Lu, Pengzhan Huang, Yinnian He. Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 815-845. doi: 10.3934/dcdsb.2020143

[15]

Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233

[16]

Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033

[17]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571

[18]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[19]

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052

[20]

Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3737-3765. doi: 10.3934/dcds.2019230

 Impact Factor: 0.263

Article outline

Figures and Tables

[Back to Top]