doi: 10.3934/era.2020089

Error estimates for second-order SAV finite element method to phase field crystal model

1. 

School of Computational Science and Electronics, Hunan Institute of Engineering, Xiangtan 411104, Hunan, China

2. 

School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and, Engineering, Xiangtan University, Xiangtan 411105, Hunan, China

* Corresponding author, Liupeng Wang

Received  March 2020 Revised  July 2020 Published  August 2020

Fund Project: The first author is supported by the General Project Hunan Provincial Education Department of China (19C0467)

In this paper, the second-order scalar auxiliary variable approach in time and linear finite element method in space are employed for solving the Cahn-Hilliard type equation of the phase field crystal model. The energy stability of the fully discrete scheme and the boundedness of numerical solution are studied. The rigorous error estimates of order $ O(\tau^2+h^2) $ in the sense of $ L^2 $-norm is derived. Finally, some numerical results are given to demonstrate the theoretical analysis.

Citation: Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, doi: 10.3934/era.2020089
References:
[1]

A. J. Archer, D. J. Ratliff, A. M. Rucklidge and P. Subramanian, Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations, Phys. Rev. E, 100 (2019). doi: 10.1103/PhysRevE.100.022140.  Google Scholar

[2]

A. BaskaranJ. S. LowengrubC. Wang and S. M. Wise, Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 51 (2013), 2851-2873.  doi: 10.1137/120880677.  Google Scholar

[3]

S. C. Brenner, $C^0$ interior penalty methods, in Frontiers in Numerical Analysis, Lect. Notes Comput. Sci. Eng., 85, Springer, Heidelberg, 2012, 79–147. doi: 10.1007/978-3-642-23914-4_2.  Google Scholar

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M. Cheng and J. A. Warren, An efficient algorithm for solving the phase field crystal model, J. Comput. Phys., 227 (2008), 6241-6248.  doi: 10.1016/j.jcp.2008.03.012.  Google Scholar

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K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002). doi: 10.1103/PhysRevLett.88.245701.  Google Scholar

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K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E, 70 (2004). doi: 10.1103/PhysRevE.70.051605.  Google Scholar

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K. R. Elder, N. Provatas, J. Berry, P. Stefanovic and M. Grant, Phase-field crystal modeling and classical density functional theory of freezing, Phys. Rev. B, 75 (2007). doi: 10.1103/PhysRevB.75.064107.  Google Scholar

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D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, unpublished article, (1998), 1–15. Google Scholar

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H. Gomez and X. Nogueira, An unconditionally energy-stable method for the phase field crystal equation, Comput. Methods Appl. Mech. Engrg., 249/252 (2012), 52-61.  doi: 10.1016/j.cma.2012.03.002.  Google Scholar

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M. Grasselli and M. Pierre, Energy stable and convergent finite element schemes for the modified phase field crystal equation, ESAIM Math. Model. Numer. Anal., 50 (2016), 1523-1560.  doi: 10.1051/m2an/2015092.  Google Scholar

[11]

R. Guo and Y. Xu, Local discontinuous Galerkin method and high order semi-implicit scheme for the phase field crystal equation, SIAM J. Sci. Comput., 38 (2016), A105–A127. doi: 10.1137/15M1038803.  Google Scholar

[12]

R. Guo and Y. Xu, A high order adaptive time-stepping strategy and local discontinuous Galerkin method for the modified phase field crystal equation, Commun. Comput. Phys., 24 (2018), 123-151.  doi: 10.4208/cicp.OA-2017-0074.  Google Scholar

[13]

Z. HuS. M. WiseC. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339.  doi: 10.1016/j.jcp.2009.04.020.  Google Scholar

[14]

K. Jiang and P. Zhang, Numerical methods for quasicrystals, J. Comput. Phys., 256 (2014), 428-440.  doi: 10.1016/j.jcp.2013.08.034.  Google Scholar

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X. Jing and Q. Wang, Linear second order energy stable schemes for phase field crystal growth models with nonlocal constraints, Comput. Math. Appl., 79 (2020), 764-788.  doi: 10.1016/j.camwa.2019.07.030.  Google Scholar

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J. Kim and J. Shin, An unconditionally gradient stable numerical method for the Ohta-Kawasaki model, Bull. Korean Math. Soc., 54 (2017), 145-158.  doi: 10.4134/BKMS.b150980.  Google Scholar

[17]

X. Li and J. Shen, Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation, Adv. Comput. Math., 46 (2020), 20pp. doi: 10.1007/s10444-020-09789-9.  Google Scholar

[18]

Z. Liu and X. Li, Two fast and efficient linear semi-implicit approaches with unconditional energy stability for nonlocal phase field crystal equation, Appl. Numer. Math., 150 (2020), 491-506.  doi: 10.1016/j.apnum.2019.10.019.  Google Scholar

[19]

S. PeiY. Hou and and B. You, A linearly second-order energy stable scheme for the phase field crystal model, Appl. Numer. Math., 140 (2019), 134-164.  doi: 10.1016/j.apnum.2019.01.017.  Google Scholar

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S. Praetorius, Efficient Solvers for the Phase-Field Crystal Equation, Ph.D dissertation, Technischen Universität Dresden, 2015. Google Scholar

[21]

N. ProvatasJ. A. DantzigB. AthreyaP. ChanP. StefanovicN. Goldenfeld and K. R. Elder, Using the phase-field crystal method in the multi-scale modeling of microstructure evolution, JOM, 59 (2007), 83-90.  doi: 10.1007/s11837-007-0095-3.  Google Scholar

[22]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.  Google Scholar

[23]

J. ShenJ. Xu and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353 (2018), 407-416.  doi: 10.1016/j.jcp.2017.10.021.  Google Scholar

[24]

J. Shen and J. Xu, Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows, SIAM J. Numer. Anal., 56 (2018), 2895-2912.  doi: 10.1137/17M1159968.  Google Scholar

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V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-33122-0.  Google Scholar

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P. VignalL. DalcinD. L. BrownN. Collier and V. M. Calo, An energy-stable convex splitting for the phase-field crystal equation, Comput. Struct., 158 (2015), 355-368.  doi: 10.1016/j.compstruc.2015.05.029.  Google Scholar

[27]

C. Wang and S. M. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49 (2011), 945-969.  doi: 10.1137/090752675.  Google Scholar

[28]

L. WangY. Huang and K. Jiang, Error analysis of SAV finite element method to phase field crystal model, Numer. Math. Theor. Meth. Appl., 13 (2020), 372-399.  doi: 10.4208/nmtma.oa-2019-0110.  Google Scholar

[29]

S. M. Wise, C. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal. 47 (2009), 2269–2288. doi: 10.1137/080738143.  Google Scholar

[30]

C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal., 44 (2006), 1759-1779.  doi: 10.1137/050628143.  Google Scholar

[31]

X. Yang and D. Han, Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model, J. Comput. Phys., 330 (2017), 1116-1134.  doi: 10.1016/j.jcp.2016.10.020.  Google Scholar

show all references

References:
[1]

A. J. Archer, D. J. Ratliff, A. M. Rucklidge and P. Subramanian, Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations, Phys. Rev. E, 100 (2019). doi: 10.1103/PhysRevE.100.022140.  Google Scholar

[2]

A. BaskaranJ. S. LowengrubC. Wang and S. M. Wise, Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 51 (2013), 2851-2873.  doi: 10.1137/120880677.  Google Scholar

[3]

S. C. Brenner, $C^0$ interior penalty methods, in Frontiers in Numerical Analysis, Lect. Notes Comput. Sci. Eng., 85, Springer, Heidelberg, 2012, 79–147. doi: 10.1007/978-3-642-23914-4_2.  Google Scholar

[4]

M. Cheng and J. A. Warren, An efficient algorithm for solving the phase field crystal model, J. Comput. Phys., 227 (2008), 6241-6248.  doi: 10.1016/j.jcp.2008.03.012.  Google Scholar

[5]

K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002). doi: 10.1103/PhysRevLett.88.245701.  Google Scholar

[6]

K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E, 70 (2004). doi: 10.1103/PhysRevE.70.051605.  Google Scholar

[7]

K. R. Elder, N. Provatas, J. Berry, P. Stefanovic and M. Grant, Phase-field crystal modeling and classical density functional theory of freezing, Phys. Rev. B, 75 (2007). doi: 10.1103/PhysRevB.75.064107.  Google Scholar

[8]

D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, unpublished article, (1998), 1–15. Google Scholar

[9]

H. Gomez and X. Nogueira, An unconditionally energy-stable method for the phase field crystal equation, Comput. Methods Appl. Mech. Engrg., 249/252 (2012), 52-61.  doi: 10.1016/j.cma.2012.03.002.  Google Scholar

[10]

M. Grasselli and M. Pierre, Energy stable and convergent finite element schemes for the modified phase field crystal equation, ESAIM Math. Model. Numer. Anal., 50 (2016), 1523-1560.  doi: 10.1051/m2an/2015092.  Google Scholar

[11]

R. Guo and Y. Xu, Local discontinuous Galerkin method and high order semi-implicit scheme for the phase field crystal equation, SIAM J. Sci. Comput., 38 (2016), A105–A127. doi: 10.1137/15M1038803.  Google Scholar

[12]

R. Guo and Y. Xu, A high order adaptive time-stepping strategy and local discontinuous Galerkin method for the modified phase field crystal equation, Commun. Comput. Phys., 24 (2018), 123-151.  doi: 10.4208/cicp.OA-2017-0074.  Google Scholar

[13]

Z. HuS. M. WiseC. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339.  doi: 10.1016/j.jcp.2009.04.020.  Google Scholar

[14]

K. Jiang and P. Zhang, Numerical methods for quasicrystals, J. Comput. Phys., 256 (2014), 428-440.  doi: 10.1016/j.jcp.2013.08.034.  Google Scholar

[15]

X. Jing and Q. Wang, Linear second order energy stable schemes for phase field crystal growth models with nonlocal constraints, Comput. Math. Appl., 79 (2020), 764-788.  doi: 10.1016/j.camwa.2019.07.030.  Google Scholar

[16]

J. Kim and J. Shin, An unconditionally gradient stable numerical method for the Ohta-Kawasaki model, Bull. Korean Math. Soc., 54 (2017), 145-158.  doi: 10.4134/BKMS.b150980.  Google Scholar

[17]

X. Li and J. Shen, Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation, Adv. Comput. Math., 46 (2020), 20pp. doi: 10.1007/s10444-020-09789-9.  Google Scholar

[18]

Z. Liu and X. Li, Two fast and efficient linear semi-implicit approaches with unconditional energy stability for nonlocal phase field crystal equation, Appl. Numer. Math., 150 (2020), 491-506.  doi: 10.1016/j.apnum.2019.10.019.  Google Scholar

[19]

S. PeiY. Hou and and B. You, A linearly second-order energy stable scheme for the phase field crystal model, Appl. Numer. Math., 140 (2019), 134-164.  doi: 10.1016/j.apnum.2019.01.017.  Google Scholar

[20]

S. Praetorius, Efficient Solvers for the Phase-Field Crystal Equation, Ph.D dissertation, Technischen Universität Dresden, 2015. Google Scholar

[21]

N. ProvatasJ. A. DantzigB. AthreyaP. ChanP. StefanovicN. Goldenfeld and K. R. Elder, Using the phase-field crystal method in the multi-scale modeling of microstructure evolution, JOM, 59 (2007), 83-90.  doi: 10.1007/s11837-007-0095-3.  Google Scholar

[22]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.  Google Scholar

[23]

J. ShenJ. Xu and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353 (2018), 407-416.  doi: 10.1016/j.jcp.2017.10.021.  Google Scholar

[24]

J. Shen and J. Xu, Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows, SIAM J. Numer. Anal., 56 (2018), 2895-2912.  doi: 10.1137/17M1159968.  Google Scholar

[25]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-33122-0.  Google Scholar

[26]

P. VignalL. DalcinD. L. BrownN. Collier and V. M. Calo, An energy-stable convex splitting for the phase-field crystal equation, Comput. Struct., 158 (2015), 355-368.  doi: 10.1016/j.compstruc.2015.05.029.  Google Scholar

[27]

C. Wang and S. M. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49 (2011), 945-969.  doi: 10.1137/090752675.  Google Scholar

[28]

L. WangY. Huang and K. Jiang, Error analysis of SAV finite element method to phase field crystal model, Numer. Math. Theor. Meth. Appl., 13 (2020), 372-399.  doi: 10.4208/nmtma.oa-2019-0110.  Google Scholar

[29]

S. M. Wise, C. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal. 47 (2009), 2269–2288. doi: 10.1137/080738143.  Google Scholar

[30]

C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal., 44 (2006), 1759-1779.  doi: 10.1137/050628143.  Google Scholar

[31]

X. Yang and D. Han, Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model, J. Comput. Phys., 330 (2017), 1116-1134.  doi: 10.1016/j.jcp.2016.10.020.  Google Scholar

Figure 1.  The phase evolution occurs
Figure 2.  The energy dissipative occurs
Figure 3.  The energy energy changing processes with difference $ \tau $
Table 1.  Time errors and convergence rates
Coarse $ \tau $ Fine $ \tau $ $ \|e_{\psi}\| $ rate $ \|e_{\phi}\| $ rate $ |e_{s}| $ rate
$ 2^{-10} $ $ 2^{-11} $ $ 1.13\text{E-}6 $ $ – -$ $ 3.72\text{E-}7 $ $ – -$ $ 1.86\text{E-}8 $ $ – -$
$ 2^{-11} $ $ 2^{-12} $ $ 2.82\text{E-}7 $ $ 2.01 $ $ 9.26\text{E-}8 $ $ 2.00 $ $ 5.74\text{E-}9 $ $ 1.70 $
$ 2^{-12} $ $ 2^{-13} $ $ 7.02\text{E-}8 $ $ 2.00 $ $ 2.31\text{E-}6 $ $ 2.00 $ $ 1.62\text{E-}9 $ $ 1.83 $
$ 2^{-13} $ $ 2^{-14} $ $ 1.75\text{E-}8 $ $ 2.00 $ $ 5.78\text{E-}9 $ $ 2.00 $ $ 4.30\text{E-}10 $ $ 1.91 $
$ 2^{-14} $ $ 2^{-15} $ $ 4.38\text{E-}9 $ $ 2.00 $ $ 1.44\text{E-}9 $ $ 2.00 $ $ 1.11\text{E-}10 $ $ 1.95 $
Coarse $ \tau $ Fine $ \tau $ $ \|e_{\psi}\| $ rate $ \|e_{\phi}\| $ rate $ |e_{s}| $ rate
$ 2^{-10} $ $ 2^{-11} $ $ 1.13\text{E-}6 $ $ – -$ $ 3.72\text{E-}7 $ $ – -$ $ 1.86\text{E-}8 $ $ – -$
$ 2^{-11} $ $ 2^{-12} $ $ 2.82\text{E-}7 $ $ 2.01 $ $ 9.26\text{E-}8 $ $ 2.00 $ $ 5.74\text{E-}9 $ $ 1.70 $
$ 2^{-12} $ $ 2^{-13} $ $ 7.02\text{E-}8 $ $ 2.00 $ $ 2.31\text{E-}6 $ $ 2.00 $ $ 1.62\text{E-}9 $ $ 1.83 $
$ 2^{-13} $ $ 2^{-14} $ $ 1.75\text{E-}8 $ $ 2.00 $ $ 5.78\text{E-}9 $ $ 2.00 $ $ 4.30\text{E-}10 $ $ 1.91 $
$ 2^{-14} $ $ 2^{-15} $ $ 4.38\text{E-}9 $ $ 2.00 $ $ 1.44\text{E-}9 $ $ 2.00 $ $ 1.11\text{E-}10 $ $ 1.95 $
Table 2.  Space errors and convergence rates
Coarse $ h $ Fine $ h $ $ \|e_{\psi}\| $ rate $ \|e_{\phi}\| $ rate $ |e_{s}| $ rate
$ 2^{-4}L_1 $ $ 2^{-5}L_1 $ $ 9.42\text{E-}2 $ $ -- $ $ 2.87\text{E-}4 $ $ -- $ $ 1.63\text{E-}2 $ $ -- $
$ 2^{-5}L_1 $ $ 2^{-6}L_1 $ $ 2.41\text{E-}2 $ $ 1.97 $ $ 7.19\text{E-}5 $ $ 2.06 $ $ 4.26\text{E-}3 $ $ 1.94 $
$ 2^{-6}L_1 $ $ 2^{-7}L_1 $ $ 6.05\text{E-}3 $ $ 1.99 $ $ 1.84\text{E-}5 $ $ 1.97 $ $ 1.08\text{E-}3 $ $ 1.98 $
$ 2^{-7}L_1 $ $ 2^{-8}L_1 $ $ 1.51\text{E-}3 $ $ 2.00 $ $ 4.67\text{E-}6 $ $ 1.98 $ $ 2.70\text{E-}4 $ $ 2.00 $
$ 2^{-8}L_1 $ $ 2^{-9}L_1 $ $ 3.78\text{E-}4 $ $ 2.00 $ $ 1.17\text{E-}6 $ $ 2.00 $ $ 6.75\text{E-}7 $ $ 2.00 $
Coarse $ h $ Fine $ h $ $ \|e_{\psi}\| $ rate $ \|e_{\phi}\| $ rate $ |e_{s}| $ rate
$ 2^{-4}L_1 $ $ 2^{-5}L_1 $ $ 9.42\text{E-}2 $ $ -- $ $ 2.87\text{E-}4 $ $ -- $ $ 1.63\text{E-}2 $ $ -- $
$ 2^{-5}L_1 $ $ 2^{-6}L_1 $ $ 2.41\text{E-}2 $ $ 1.97 $ $ 7.19\text{E-}5 $ $ 2.06 $ $ 4.26\text{E-}3 $ $ 1.94 $
$ 2^{-6}L_1 $ $ 2^{-7}L_1 $ $ 6.05\text{E-}3 $ $ 1.99 $ $ 1.84\text{E-}5 $ $ 1.97 $ $ 1.08\text{E-}3 $ $ 1.98 $
$ 2^{-7}L_1 $ $ 2^{-8}L_1 $ $ 1.51\text{E-}3 $ $ 2.00 $ $ 4.67\text{E-}6 $ $ 1.98 $ $ 2.70\text{E-}4 $ $ 2.00 $
$ 2^{-8}L_1 $ $ 2^{-9}L_1 $ $ 3.78\text{E-}4 $ $ 2.00 $ $ 1.17\text{E-}6 $ $ 2.00 $ $ 6.75\text{E-}7 $ $ 2.00 $
Table 3.  Time errors and convergence rates
Coarse $ \tau $ Fine $ \tau $ $ \|e_{\psi}\| $ rate $ \|e_{\phi}\| $ rate $ |e_{s}| $ rate
$ 2^{-7} $ $ 2^{-8} $ $ 8.06\text{E-}5 $ $ – -$ $ 4.84\text{E-}5 $ $ – -$ $ 3.79\text{E-}6 $ $ – -$
$ 2^{-8} $ $ 2^{-9} $ $ 2.19\text{E-}5 $ $ 1.88 $ $ 1.25\text{E-}5 $ $ 1.95 $ $ 1.03\text{E-}6 $ $ 1.97 $
$ 2^{-9} $ $ 2^{-10} $ $ 5.72\text{E-}6 $ $ 1.93 $ $ 3.21\text{E-}6 $ $ 1.97 $ $ 2.62\text{E-}7 $ $ 1.97 $
$ 2^{-10} $ $ 2^{-11} $ $ 1.47\text{E-}6 $ $ 1.96 $ $ 8.19\text{E-}7 $ $ 1.89 $ $ 6.68\text{E-}8 $ $ 1.97 $
$ 2^{-11} $ $ 2^{-12} $ $ 3.75\text{E-}7 $ $ 1.97 $ $ 2.06\text{E-}7 $ $ 1.99 $ $ 1.69\text{E-}8 $ $ 1.98 $
Coarse $ \tau $ Fine $ \tau $ $ \|e_{\psi}\| $ rate $ \|e_{\phi}\| $ rate $ |e_{s}| $ rate
$ 2^{-7} $ $ 2^{-8} $ $ 8.06\text{E-}5 $ $ – -$ $ 4.84\text{E-}5 $ $ – -$ $ 3.79\text{E-}6 $ $ – -$
$ 2^{-8} $ $ 2^{-9} $ $ 2.19\text{E-}5 $ $ 1.88 $ $ 1.25\text{E-}5 $ $ 1.95 $ $ 1.03\text{E-}6 $ $ 1.97 $
$ 2^{-9} $ $ 2^{-10} $ $ 5.72\text{E-}6 $ $ 1.93 $ $ 3.21\text{E-}6 $ $ 1.97 $ $ 2.62\text{E-}7 $ $ 1.97 $
$ 2^{-10} $ $ 2^{-11} $ $ 1.47\text{E-}6 $ $ 1.96 $ $ 8.19\text{E-}7 $ $ 1.89 $ $ 6.68\text{E-}8 $ $ 1.97 $
$ 2^{-11} $ $ 2^{-12} $ $ 3.75\text{E-}7 $ $ 1.97 $ $ 2.06\text{E-}7 $ $ 1.99 $ $ 1.69\text{E-}8 $ $ 1.98 $
Table 4.  Space errors and convergence rates
Coarse $ h $ Fine $ h $ $ \|e_{\psi}\| $ rate $ \|e_{\phi}\| $ rate $ |e_{s}| $ rate
$ 2^{-2}L_2 $ $ 2^{-3}L_2 $ $ 4.93\text{E-}1 $ $ -- $ $ 2.72\text{E-}2 $ $ -- $ $ 1.34\text{E-}2 $ $ -- $
$ 2^{-3}L_2 $ $ 2^{-4}L_2 $ $ 1.57\text{E-}2 $ $ 1.65 $ $ 9.93\text{E-}3 $ $ 1.45 $ $ 3.94\text{E-}3 $ $ 1.77 $
$ 2^{-4}L_2 $ $ 2^{-5}L_2 $ $ 3.84\text{E-}2 $ $ 2.03 $ $ 3.29\text{E-}3 $ $ 1.59 $ $ 1.03\text{E-}3 $ $ 1.94 $
$ 2^{-5}L_2 $ $ 2^{-6}L_2 $ $ 9.43\text{E-}3 $ $ 2.02 $ $ 9.10\text{E-}4 $ $ 1.85 $ $ 2.60\text{E-}4 $ $ 1.99 $
$ 2^{-6}L_2 $ $ 2^{-7}L_2 $ $ 2.35\text{E-}2 $ $ 2.00 $ $ 2.35\text{E-}4 $ $ 1.96 $ $ 6.50\text{E-}5 $ $ 2.00 $
Coarse $ h $ Fine $ h $ $ \|e_{\psi}\| $ rate $ \|e_{\phi}\| $ rate $ |e_{s}| $ rate
$ 2^{-2}L_2 $ $ 2^{-3}L_2 $ $ 4.93\text{E-}1 $ $ -- $ $ 2.72\text{E-}2 $ $ -- $ $ 1.34\text{E-}2 $ $ -- $
$ 2^{-3}L_2 $ $ 2^{-4}L_2 $ $ 1.57\text{E-}2 $ $ 1.65 $ $ 9.93\text{E-}3 $ $ 1.45 $ $ 3.94\text{E-}3 $ $ 1.77 $
$ 2^{-4}L_2 $ $ 2^{-5}L_2 $ $ 3.84\text{E-}2 $ $ 2.03 $ $ 3.29\text{E-}3 $ $ 1.59 $ $ 1.03\text{E-}3 $ $ 1.94 $
$ 2^{-5}L_2 $ $ 2^{-6}L_2 $ $ 9.43\text{E-}3 $ $ 2.02 $ $ 9.10\text{E-}4 $ $ 1.85 $ $ 2.60\text{E-}4 $ $ 1.99 $
$ 2^{-6}L_2 $ $ 2^{-7}L_2 $ $ 2.35\text{E-}2 $ $ 2.00 $ $ 2.35\text{E-}4 $ $ 1.96 $ $ 6.50\text{E-}5 $ $ 2.00 $
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