# 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  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] Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 [2] 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 [3] 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  doi: 10.3934/mcrf.2021014 [4] Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 [5] Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011 [6] Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035 [7] Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265 [8] Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329 [9] Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200 [10] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [11] Lars Grüne, Luca Mechelli, Simon Pirkelmann, Stefan Volkwein. Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021013 [12] Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021  doi: 10.3934/fods.2021005 [13] Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451 [14] Todd Hurst, Volker Rehbock. Optimizing micro-algae production in a raceway pond with variable depth. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021027 [15] Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597 [16] Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021006 [17] Charles Amorim, Miguel Loayza, Marko A. Rojas-Medar. The nonstationary flows of micropolar fluids with thermal convection: An iterative approach. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2509-2535. doi: 10.3934/dcdsb.2020193 [18] Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 [19] Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637 [20] Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

Impact Factor: 0.263