# American Institute of Mathematical Sciences

March  2021, 29(1): 1735-1752. 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  March 2021 Early access  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, 2021, 29 (1) : 1735-1752. 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. Baskaran, J. S. Lowengrub, C. 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. Hu, S. M. Wise, C. 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. Pei, Y. 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. Provatas, J. A. Dantzig, B. Athreya, P. Chan, P. Stefanovic, N. 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. Shen, J. 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. Vignal, L. Dalcin, D. L. Brown, N. 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. Wang, Y. 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. Baskaran, J. S. Lowengrub, C. 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. Hu, S. M. Wise, C. 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. Pei, Y. 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. Provatas, J. A. Dantzig, B. Athreya, P. Chan, P. Stefanovic, N. 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. Shen, J. 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. Vignal, L. Dalcin, D. L. Brown, N. 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. Wang, Y. 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
The phase evolution occurs
The energy dissipative occurs
The energy energy changing processes with difference $\tau$
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 $2^{-10}$ $2^{-11}$ $1.13\text{E-}6$ $– -$ $3.72\text{E-}7$ $– -$ $1.86\text{E-}8$ $– -$ $2^{-11}$ $2^{-12}$ $2.82\text{E-}7$ $2.01$ $9.26\text{E-}8$ $2.00$ $5.74\text{E-}9$ $1.70$ $2^{-12}$ $2^{-13}$ $7.02\text{E-}8$ $2.00$ $2.31\text{E-}6$ $2.00$ $1.62\text{E-}9$ $1.83$ $2^{-13}$ $2^{-14}$ $1.75\text{E-}8$ $2.00$ $5.78\text{E-}9$ $2.00$ $4.30\text{E-}10$ $1.91$ $2^{-14}$ $2^{-15}$ $4.38\text{E-}9$ $2.00$ $1.44\text{E-}9$ $2.00$ $1.11\text{E-}10$ $1.95$
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 $2^{-4}L_1$ $2^{-5}L_1$ $9.42\text{E-}2$ $--$ $2.87\text{E-}4$ $--$ $1.63\text{E-}2$ $--$ $2^{-5}L_1$ $2^{-6}L_1$ $2.41\text{E-}2$ $1.97$ $7.19\text{E-}5$ $2.06$ $4.26\text{E-}3$ $1.94$ $2^{-6}L_1$ $2^{-7}L_1$ $6.05\text{E-}3$ $1.99$ $1.84\text{E-}5$ $1.97$ $1.08\text{E-}3$ $1.98$ $2^{-7}L_1$ $2^{-8}L_1$ $1.51\text{E-}3$ $2.00$ $4.67\text{E-}6$ $1.98$ $2.70\text{E-}4$ $2.00$ $2^{-8}L_1$ $2^{-9}L_1$ $3.78\text{E-}4$ $2.00$ $1.17\text{E-}6$ $2.00$ $6.75\text{E-}7$ $2.00$
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 $2^{-7}$ $2^{-8}$ $8.06\text{E-}5$ $– -$ $4.84\text{E-}5$ $– -$ $3.79\text{E-}6$ $– -$ $2^{-8}$ $2^{-9}$ $2.19\text{E-}5$ $1.88$ $1.25\text{E-}5$ $1.95$ $1.03\text{E-}6$ $1.97$ $2^{-9}$ $2^{-10}$ $5.72\text{E-}6$ $1.93$ $3.21\text{E-}6$ $1.97$ $2.62\text{E-}7$ $1.97$ $2^{-10}$ $2^{-11}$ $1.47\text{E-}6$ $1.96$ $8.19\text{E-}7$ $1.89$ $6.68\text{E-}8$ $1.97$ $2^{-11}$ $2^{-12}$ $3.75\text{E-}7$ $1.97$ $2.06\text{E-}7$ $1.99$ $1.69\text{E-}8$ $1.98$
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 $2^{-2}L_2$ $2^{-3}L_2$ $4.93\text{E-}1$ $--$ $2.72\text{E-}2$ $--$ $1.34\text{E-}2$ $--$ $2^{-3}L_2$ $2^{-4}L_2$ $1.57\text{E-}2$ $1.65$ $9.93\text{E-}3$ $1.45$ $3.94\text{E-}3$ $1.77$ $2^{-4}L_2$ $2^{-5}L_2$ $3.84\text{E-}2$ $2.03$ $3.29\text{E-}3$ $1.59$ $1.03\text{E-}3$ $1.94$ $2^{-5}L_2$ $2^{-6}L_2$ $9.43\text{E-}3$ $2.02$ $9.10\text{E-}4$ $1.85$ $2.60\text{E-}4$ $1.99$ $2^{-6}L_2$ $2^{-7}L_2$ $2.35\text{E-}2$ $2.00$ $2.35\text{E-}4$ $1.96$ $6.50\text{E-}5$ $2.00$
 [1] Yingwen Guo, Yinnian He. Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2583-2609. doi: 10.3934/dcdsb.2015.20.2583 [2] Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873 [3] Sondre Tesdal Galtung. A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1243-1268. doi: 10.3934/dcds.2018051 [4] Yoshiho Akagawa, Elliott Ginder, Syota Koide, Seiro Omata, Karel Svadlenka. A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021153 [5] Yi Shi, Kai Bao, Xiao-Ping Wang. 3D adaptive finite element method for a phase field model for the moving contact line problems. Inverse Problems & Imaging, 2013, 7 (3) : 947-959. doi: 10.3934/ipi.2013.7.947 [6] Kai Jiang, Wei Si. High-order energy stable schemes of incommensurate phase-field crystal model. Electronic Research Archive, 2020, 28 (2) : 1077-1093. doi: 10.3934/era.2020059 [7] Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641 [8] Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663 [9] Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024 [10] Mouhamadou Samsidy Goudiaby, Ababacar Diagne, Leon Matar Tine. Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3499-3514. doi: 10.3934/cpaa.2021116 [11] Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control & Related Fields, 2020, 10 (2) : 333-363. doi: 10.3934/mcrf.2019041 [12] Alexander Zlotnik. The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis. Kinetic & Related Models, 2015, 8 (3) : 587-613. doi: 10.3934/krm.2015.8.587 [13] Anouar El Harrak, Hatim Tayeq, Amal Bergam. A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062 [14] Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 [15] Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 37-54. doi: 10.3934/dcdss.2017002 [16] Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017 [17] Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044 [18] Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014 [19] 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 [20] Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040

2020 Impact Factor: 1.833

## Metrics

• HTML views (474)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar