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On the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures and logarithmic potentials
1. | Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France |
2. | Xiamen University, School of Mathematical Sciences, Fujian Provincial Key Laboratory of Mathematical Modeling, and High Performance Scientific Computing, Xiamen, Fujian, China |
3. | Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France |
4. | Université de Picardie Jules Verne, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, UMR CNRS 7352, Pôle Scientifique, 33, rue Saint Leu, F-80039 Amiens, France |
Our aim in this article is to study generalizations of the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the system. In particular, we prove the existence of the global attractor. Furthermore, we give some numerical simulations, obtained with the $\mathtt{FreeFem++}$ software [
References:
[1] |
S. Aizicovici and E. Feireisl,
Long-time stabilization of solutions to a phase-field model with memory, J. Evol. Eqns., 1 (2001), 69-84.
doi: 10.1007/PL00001365. |
[2] |
S. Aizicovici, E. Feireisl and F. Issard-Roch,
Long-time convergence of solutions to a phase-field system, Math. Methods Appl. Sci., 24 (2001), 277-287.
doi: 10.1002/mma.215. |
[3] |
D. Brochet, X. Chen and D. Hilhorst,
Finite dimensional exponential attractors for the phase-field model, Appl. Anal., 49 (1993), 197-212.
doi: 10.1080/00036819108840173. |
[4] |
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[5] |
G. Caginalp,
An analysis of a phase-field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.
doi: 10.1007/BF00254827. |
[6] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system Ⅰ. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267.
|
[7] |
P. J. Chen and M. E. Gurtin,
On a theory of heat involving two temperatures, J. Appl. Math. Phys. (ZAMP), 19 (1968), 614-627.
|
[8] |
P. J. Chen, M. E. Gurtin and W. O. Williams,
A note on a non-simple heat conduction, J. Appl. Math. Phys. (ZAMP), 19 (1968), 969-970.
doi: 10.1007/BF01602278. |
[9] |
P. J. Chen, M. E. Gurtin and W. O. Williams,
On the thermodynamics of non-simple materials with two temperatures, J. Appl. Math. Phys. (ZAMP), 20 (1969), 107-112.
doi: 10.1007/BF01591120. |
[10] |
L. Cherfils and A. Miranville,
Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129.
|
[11] |
L. Cherfils and A. Miranville,
On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.
doi: 10.1007/s10492-009-0008-6. |
[12] |
L. Cherfils, S. Gatti and A. Miranville,
A doubly nonlinear parabolic equation with a singular potential, Discrete Contin. Dyn. Systems S, 4 (2011), 51-66.
doi: 10.3934/dcdss.2011.4.51. |
[13] |
R. Chill, E Fasangovà and J. Prüss,
Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462.
doi: 10.1002/mana.200410431. |
[14] |
C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving medis, Phys. Review Letters, 94 (2005), 154301. |
[15] |
B. Doumbé, Etude de modèles de champs de phase de type Caginalp, Université de Poitiers, 2013. |
[16] |
A. S. El-Karamany and M. A. Ezzat,
On the two-temperature Green-Naghdi thermoelasticity theories, J. Thermal Stresses, 34 (2011), 1207-1226.
doi: 10.1080/01495739.2011.608313. |
[17] |
C. G. Gal and M. Grasselli,
The nonisothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Systems A, 22 (2008), 1009-1040.
doi: 10.3934/dcds.2008.22.1009. |
[18] |
S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in Differential Equations: Inverse and Direct Problems (Proceedings of the Workshop "Evolutiob Equations: Inverse and Direct Problems", Cortona, June 21-25, 2004), A series of Lecture notes in pure and applied mathematics, 251, A. Favini and A. Lorenzi eds., Chapman & Hall, 2006,149–170.
doi: 10.1201/9781420011135.ch9. |
[19] |
M. Grasselli, A. Miranville, V. Pata and S. Zelik,
Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280 (2007), 1475-1509.
doi: 10.1002/mana.200510560. |
[20] |
M. Grasselli, A. Miranville and G. Schimperna,
The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Systems A, 28 (2010), 67-98.
doi: 10.3934/dcds.2010.28.67. |
[21] |
M. Grasselli and V. Pata,
Existence of a universal attractor for a fully hyperbolic phase-field system, J. Evol. Eqns., 4 (2004), 27-51.
doi: 10.1007/s00028-003-0074-2. |
[22] |
M. Grasselli, H. Petzeltová and G. Schimperna,
Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72.
doi: 10.4171/ZAA/1277. |
[23] |
A. E. Green and P. M. Naghdi,
A re-examination of the basic postulates of thermomechanics, Proc. Royal Society London A, 432 (1991), 171-194.
doi: 10.1098/rspa.1991.0012. |
[24] |
F. Hecht, O. Pironneau, A. Le Hyaric and K. Ohtsuka, Freefem++ Manual, 2012. |
[25] |
J. Jiang,
Convergence to equilibrium for a parabolic-hyperbolic phase-field model with Cattaneo heat flux law, J. Math. Anal. Appl., 341 (2008), 149-169.
doi: 10.1016/j.jmaa.2007.09.041. |
[26] |
J. Jiang,
Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law, Math. Methods Appl. Sci., 32 (2009), 1156-1182.
doi: 10.1002/mma.1092. |
[27] |
A. Miranville,
Some mathematical models in phase transition, Discrete Contin. Dyn. Systems S, 7 (2014), 271-306.
doi: 10.3934/dcdss.2014.7.271. |
[28] |
A. Miranville and R. Quintanilla,
A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal. TMA, 71 (2009), 2278-2290.
doi: 10.1016/j.na.2009.01.061. |
[29] |
A. Miranville and R. Quintanilla,
Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894.
doi: 10.1080/00036810903042182. |
[30] |
A. Miranville and R. Quintanilla,
A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150.
doi: 10.1007/s00245-010-9114-9. |
[31] |
A. Miranville and R. Quintanilla,
A type $\rm III$ phase-field system with a logarithmic potential, Appl. Math. Letters, 24 (2011), 1003-1008.
doi: 10.1016/j.aml.2011.01.016. |
[32] |
A. Miranville and R. Quintanilla,
A generalization of the Allen-Cahn equation, IMA J. Appl. Math., 80 (2015), 410-430.
doi: 10.1093/imamat/hxt044. |
[33] |
A. Miranville and R. Quintanilla,
A Caginalp phase-field system based on type Ⅲ heat conduction with two temperatures, Quart. Appl. Math., 74 (2016), 375-398.
doi: 10.1090/qam/1430. |
[34] |
A. Miranville and R. Quintanilla,
On the Caginalp phase-field systems with two temperatures and the Maxwell-Cattaneo law, Math. Methods Appl. Sci., 39 (2016), 4385-4397.
doi: 10.1002/mma.3867. |
[35] |
A. Miranville and S. Zelik,
Robust exponential attractors for singularly perturbed phase-field type equations, Electronic J. Diff. Eqns., 2002 (2002), 1-28.
|
[36] |
A. Miranville and S. Zelik,
Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[37] |
R. Quintanilla,
A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278.
doi: 10.1080/01495730903310599. |
[38] |
G. Sadaka,
Solution of 2D Boussinesq systems with FreeFem++: The flat bottom case, J. Numer. Math., 20 (2012), 303-324.
doi: 10.1515/jnum-2012-0016. |
[39] |
H. M. Youssef,
Theory of two-temperature-generalized thermoelasticity, IMA J. Appl. Math., 71 (2006), 383-390.
doi: 10.1093/imamat/hxh101. |
[40] |
Z. Zhang,
Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Comm. Pure Appl. Anal., 4 (2005), 683-693.
doi: 10.3934/cpaa.2005.4.683. |
show all references
References:
[1] |
S. Aizicovici and E. Feireisl,
Long-time stabilization of solutions to a phase-field model with memory, J. Evol. Eqns., 1 (2001), 69-84.
doi: 10.1007/PL00001365. |
[2] |
S. Aizicovici, E. Feireisl and F. Issard-Roch,
Long-time convergence of solutions to a phase-field system, Math. Methods Appl. Sci., 24 (2001), 277-287.
doi: 10.1002/mma.215. |
[3] |
D. Brochet, X. Chen and D. Hilhorst,
Finite dimensional exponential attractors for the phase-field model, Appl. Anal., 49 (1993), 197-212.
doi: 10.1080/00036819108840173. |
[4] |
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[5] |
G. Caginalp,
An analysis of a phase-field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.
doi: 10.1007/BF00254827. |
[6] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system Ⅰ. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267.
|
[7] |
P. J. Chen and M. E. Gurtin,
On a theory of heat involving two temperatures, J. Appl. Math. Phys. (ZAMP), 19 (1968), 614-627.
|
[8] |
P. J. Chen, M. E. Gurtin and W. O. Williams,
A note on a non-simple heat conduction, J. Appl. Math. Phys. (ZAMP), 19 (1968), 969-970.
doi: 10.1007/BF01602278. |
[9] |
P. J. Chen, M. E. Gurtin and W. O. Williams,
On the thermodynamics of non-simple materials with two temperatures, J. Appl. Math. Phys. (ZAMP), 20 (1969), 107-112.
doi: 10.1007/BF01591120. |
[10] |
L. Cherfils and A. Miranville,
Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129.
|
[11] |
L. Cherfils and A. Miranville,
On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.
doi: 10.1007/s10492-009-0008-6. |
[12] |
L. Cherfils, S. Gatti and A. Miranville,
A doubly nonlinear parabolic equation with a singular potential, Discrete Contin. Dyn. Systems S, 4 (2011), 51-66.
doi: 10.3934/dcdss.2011.4.51. |
[13] |
R. Chill, E Fasangovà and J. Prüss,
Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462.
doi: 10.1002/mana.200410431. |
[14] |
C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving medis, Phys. Review Letters, 94 (2005), 154301. |
[15] |
B. Doumbé, Etude de modèles de champs de phase de type Caginalp, Université de Poitiers, 2013. |
[16] |
A. S. El-Karamany and M. A. Ezzat,
On the two-temperature Green-Naghdi thermoelasticity theories, J. Thermal Stresses, 34 (2011), 1207-1226.
doi: 10.1080/01495739.2011.608313. |
[17] |
C. G. Gal and M. Grasselli,
The nonisothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Systems A, 22 (2008), 1009-1040.
doi: 10.3934/dcds.2008.22.1009. |
[18] |
S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in Differential Equations: Inverse and Direct Problems (Proceedings of the Workshop "Evolutiob Equations: Inverse and Direct Problems", Cortona, June 21-25, 2004), A series of Lecture notes in pure and applied mathematics, 251, A. Favini and A. Lorenzi eds., Chapman & Hall, 2006,149–170.
doi: 10.1201/9781420011135.ch9. |
[19] |
M. Grasselli, A. Miranville, V. Pata and S. Zelik,
Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280 (2007), 1475-1509.
doi: 10.1002/mana.200510560. |
[20] |
M. Grasselli, A. Miranville and G. Schimperna,
The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Systems A, 28 (2010), 67-98.
doi: 10.3934/dcds.2010.28.67. |
[21] |
M. Grasselli and V. Pata,
Existence of a universal attractor for a fully hyperbolic phase-field system, J. Evol. Eqns., 4 (2004), 27-51.
doi: 10.1007/s00028-003-0074-2. |
[22] |
M. Grasselli, H. Petzeltová and G. Schimperna,
Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72.
doi: 10.4171/ZAA/1277. |
[23] |
A. E. Green and P. M. Naghdi,
A re-examination of the basic postulates of thermomechanics, Proc. Royal Society London A, 432 (1991), 171-194.
doi: 10.1098/rspa.1991.0012. |
[24] |
F. Hecht, O. Pironneau, A. Le Hyaric and K. Ohtsuka, Freefem++ Manual, 2012. |
[25] |
J. Jiang,
Convergence to equilibrium for a parabolic-hyperbolic phase-field model with Cattaneo heat flux law, J. Math. Anal. Appl., 341 (2008), 149-169.
doi: 10.1016/j.jmaa.2007.09.041. |
[26] |
J. Jiang,
Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law, Math. Methods Appl. Sci., 32 (2009), 1156-1182.
doi: 10.1002/mma.1092. |
[27] |
A. Miranville,
Some mathematical models in phase transition, Discrete Contin. Dyn. Systems S, 7 (2014), 271-306.
doi: 10.3934/dcdss.2014.7.271. |
[28] |
A. Miranville and R. Quintanilla,
A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal. TMA, 71 (2009), 2278-2290.
doi: 10.1016/j.na.2009.01.061. |
[29] |
A. Miranville and R. Quintanilla,
Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894.
doi: 10.1080/00036810903042182. |
[30] |
A. Miranville and R. Quintanilla,
A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150.
doi: 10.1007/s00245-010-9114-9. |
[31] |
A. Miranville and R. Quintanilla,
A type $\rm III$ phase-field system with a logarithmic potential, Appl. Math. Letters, 24 (2011), 1003-1008.
doi: 10.1016/j.aml.2011.01.016. |
[32] |
A. Miranville and R. Quintanilla,
A generalization of the Allen-Cahn equation, IMA J. Appl. Math., 80 (2015), 410-430.
doi: 10.1093/imamat/hxt044. |
[33] |
A. Miranville and R. Quintanilla,
A Caginalp phase-field system based on type Ⅲ heat conduction with two temperatures, Quart. Appl. Math., 74 (2016), 375-398.
doi: 10.1090/qam/1430. |
[34] |
A. Miranville and R. Quintanilla,
On the Caginalp phase-field systems with two temperatures and the Maxwell-Cattaneo law, Math. Methods Appl. Sci., 39 (2016), 4385-4397.
doi: 10.1002/mma.3867. |
[35] |
A. Miranville and S. Zelik,
Robust exponential attractors for singularly perturbed phase-field type equations, Electronic J. Diff. Eqns., 2002 (2002), 1-28.
|
[36] |
A. Miranville and S. Zelik,
Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[37] |
R. Quintanilla,
A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278.
doi: 10.1080/01495730903310599. |
[38] |
G. Sadaka,
Solution of 2D Boussinesq systems with FreeFem++: The flat bottom case, J. Numer. Math., 20 (2012), 303-324.
doi: 10.1515/jnum-2012-0016. |
[39] |
H. M. Youssef,
Theory of two-temperature-generalized thermoelasticity, IMA J. Appl. Math., 71 (2006), 383-390.
doi: 10.1093/imamat/hxh101. |
[40] |
Z. Zhang,
Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Comm. Pure Appl. Anal., 4 (2005), 683-693.
doi: 10.3934/cpaa.2005.4.683. |







CPU time | rate | rate | rate | rate | |||||
1/1 | 00:00:02 | 0.00147 | - | 0.00124 | - | 0.0573 | - | 0.05544 | - |
1/4 | 00:00:22 | 0.00038 | 0.98 | 0.0003 | 0.98 | 0.02927 | 0.48 | 0.02828 | 0.49 |
1/16 | 00:05:58 | 9.5e-05 | 0.99 | 8.1e-05 | 0.99 | 0.01472 | 0.49 | 0.01422 | 0.49 |
1/64 | 01:50:22 | 2.4e-05 | 0.99 | 2e-05 | 0.99 | 0.00737 | 0.49 | 0.00711 | 0.49 |
1/256 | 22:06:19 | 6e-06 | 1 | 5e-06 | 0.99 | 0.00369 | 0.5 | 0.00357 | 0.49 |
CPU time | rate | rate | rate | rate | |||||
1/1 | 00:00:02 | 0.00147 | - | 0.00124 | - | 0.0573 | - | 0.05544 | - |
1/4 | 00:00:22 | 0.00038 | 0.98 | 0.0003 | 0.98 | 0.02927 | 0.48 | 0.02828 | 0.49 |
1/16 | 00:05:58 | 9.5e-05 | 0.99 | 8.1e-05 | 0.99 | 0.01472 | 0.49 | 0.01422 | 0.49 |
1/64 | 01:50:22 | 2.4e-05 | 0.99 | 2e-05 | 0.99 | 0.00737 | 0.49 | 0.00711 | 0.49 |
1/256 | 22:06:19 | 6e-06 | 1 | 5e-06 | 0.99 | 0.00369 | 0.5 | 0.00357 | 0.49 |
$10^{2}\cdot\delta t$ | CPU time | $N_{L^2}(u)$ | rate | $N_{L^2}(H)$ | rate | $N_{H^1}(u)$ | rate | $N_{H^1}(H)$ | rate |
1/1 | 00:00:02 | 0.00146 | - | 0.00124 | - | 0.05725 | - | 0.05544 | - |
1/4 | 00:00:25 | 0.00038 | 0.98 | 0.0003 | 0.98 | 0.02927 | 0.48 | 0.02828 | 0.49 |
1/16 | 00:07:44 | 9.5e-05 | 0.99 | 8.1e-05 | 0.99 | 0.01472 | 0.49 | 0.01422 | 0.49 |
1/64 | 01:52:09 | 2.4e-05 | 0.99 | 2e-05 | 0.99 | 0.00737 | 0.49 | 0.00711 | 0.49 |
1/256 | 23:06:55 | 6e-06 | 1 | 5e-06 | 0.99 | 0.00369 | 0.5 | 0.00357 | 0.49 |
$10^{2}\cdot\delta t$ | CPU time | $N_{L^2}(u)$ | rate | $N_{L^2}(H)$ | rate | $N_{H^1}(u)$ | rate | $N_{H^1}(H)$ | rate |
1/1 | 00:00:02 | 0.00146 | - | 0.00124 | - | 0.05725 | - | 0.05544 | - |
1/4 | 00:00:25 | 0.00038 | 0.98 | 0.0003 | 0.98 | 0.02927 | 0.48 | 0.02828 | 0.49 |
1/16 | 00:07:44 | 9.5e-05 | 0.99 | 8.1e-05 | 0.99 | 0.01472 | 0.49 | 0.01422 | 0.49 |
1/64 | 01:52:09 | 2.4e-05 | 0.99 | 2e-05 | 0.99 | 0.00737 | 0.49 | 0.00711 | 0.49 |
1/256 | 23:06:55 | 6e-06 | 1 | 5e-06 | 0.99 | 0.00369 | 0.5 | 0.00357 | 0.49 |
$10^{2}\cdot\delta t$ | CPU time | $N_{L^2}(u)$ | rate | $N_{L^2}(H)$ | rate | $N_{H^1}(u)$ | rate | $N_{H^1}(H)$ | rate |
1/1 | 00:00:01 | 0.00042 | - | 0.00039 | - | 0.02458 | - | 0.02397 | - |
1/4 | 00:00:22 | 0.00011 | 0.98 | 0.0001 | 0.98 | 0.01249 | 0.49 | 0.01215 | 0.49 |
1/16 | 00:07:09 | 2.7e-05 | 0.99 | 2.5e-05 | 0.99 | 0.00627 | 0.49 | 0.00611 | 0.49 |
1/64 | 01:27:20 | 7e-06 | 0.99 | 6e-06 | 0.99 | 0.00314 | 0.49 | 0.00306 | 0.49 |
1/256 | 22:16:47 | 2e-06 | 1 | 2e-06 | 0.99 | 0.00157 | 0.5 | 0.00154 | 0.49 |
$10^{2}\cdot\delta t$ | CPU time | $N_{L^2}(u)$ | rate | $N_{L^2}(H)$ | rate | $N_{H^1}(u)$ | rate | $N_{H^1}(H)$ | rate |
1/1 | 00:00:01 | 0.00042 | - | 0.00039 | - | 0.02458 | - | 0.02397 | - |
1/4 | 00:00:22 | 0.00011 | 0.98 | 0.0001 | 0.98 | 0.01249 | 0.49 | 0.01215 | 0.49 |
1/16 | 00:07:09 | 2.7e-05 | 0.99 | 2.5e-05 | 0.99 | 0.00627 | 0.49 | 0.00611 | 0.49 |
1/64 | 01:27:20 | 7e-06 | 0.99 | 6e-06 | 0.99 | 0.00314 | 0.49 | 0.00306 | 0.49 |
1/256 | 22:16:47 | 2e-06 | 1 | 2e-06 | 0.99 | 0.00157 | 0.5 | 0.00154 | 0.49 |
$10^{2}\cdot\delta t$ | CPU time | $N_{L^2}(u)$ | rate | $N_{L^2}(H)$ | rate | $N_{H^1}(u)$ | rate | $N_{H^1}(H)$ | rate |
1/1 | 00:00:01 | 0.00042 | - | 0.00039 | - | 0.02457 | - | 0.02397 | - |
1/4 | 00:00:24 | 0.00011 | 0.98 | 0.0001 | 0.98 | 0.01248 | 0.49 | 0.01215 | 0.49 |
1/16 | 00:07:41 | 2.7e-05 | 0.99 | 2.5e-05 | 0.99 | 0.00627 | 0.49 | 0.00611 | 0.49 |
1/64 | 01:26:11 | 7e-06 | 0.99 | 6e-06 | 0.99 | 0.00314 | 0.49 | 0.00306 | 0.49 |
1/256 | 23:06:30 | 2e-06 | 1 | 2e-06 | 0.99 | 0.00157 | 0.5 | 0.00154 | 0.49 |
$10^{2}\cdot\delta t$ | CPU time | $N_{L^2}(u)$ | rate | $N_{L^2}(H)$ | rate | $N_{H^1}(u)$ | rate | $N_{H^1}(H)$ | rate |
1/1 | 00:00:01 | 0.00042 | - | 0.00039 | - | 0.02457 | - | 0.02397 | - |
1/4 | 00:00:24 | 0.00011 | 0.98 | 0.0001 | 0.98 | 0.01248 | 0.49 | 0.01215 | 0.49 |
1/16 | 00:07:41 | 2.7e-05 | 0.99 | 2.5e-05 | 0.99 | 0.00627 | 0.49 | 0.00611 | 0.49 |
1/64 | 01:26:11 | 7e-06 | 0.99 | 6e-06 | 0.99 | 0.00314 | 0.49 | 0.00306 | 0.49 |
1/256 | 23:06:30 | 2e-06 | 1 | 2e-06 | 0.99 | 0.00157 | 0.5 | 0.00154 | 0.49 |
$10^{2}\cdot\delta t$ | CPU time | $N_{L^2}(u)$ | rate | $N_{L^2}(H)$ | rate | $N_{H^1}(u)$ | rate | $N_{H^1}(H)$ | rate |
1/1 | 00:00:01 | 0.00013 | - | 0.00012 | - | 0.01227 | - | 0.01186 | - |
1/4 | 00:00:22 | 3.4e-05 | 0.98 | 3.2e-05 | 0.98 | 0.00625 | 0.49 | 0.00603 | 0.49 |
1/16 | 00:07:19 | 9e-06 | 0.99 | 8e-06 | 0.99 | 0.00314 | 0.49 | 0.00303 | 0.49 |
1/64 | 01:29:47 | 2e-06 | 0.99 | 2e-06 | 0.99 | 0.00157 | 0.49 | 0.00152 | 0.49 |
1/256 | 22:23:29 | 1e-06 | 1 | 1e-06 | 0.99 | 0.00079 | 0.5 | 0.00076 | 0.49 |
$10^{2}\cdot\delta t$ | CPU time | $N_{L^2}(u)$ | rate | $N_{L^2}(H)$ | rate | $N_{H^1}(u)$ | rate | $N_{H^1}(H)$ | rate |
1/1 | 00:00:01 | 0.00013 | - | 0.00012 | - | 0.01227 | - | 0.01186 | - |
1/4 | 00:00:22 | 3.4e-05 | 0.98 | 3.2e-05 | 0.98 | 0.00625 | 0.49 | 0.00603 | 0.49 |
1/16 | 00:07:19 | 9e-06 | 0.99 | 8e-06 | 0.99 | 0.00314 | 0.49 | 0.00303 | 0.49 |
1/64 | 01:29:47 | 2e-06 | 0.99 | 2e-06 | 0.99 | 0.00157 | 0.49 | 0.00152 | 0.49 |
1/256 | 22:23:29 | 1e-06 | 1 | 1e-06 | 0.99 | 0.00079 | 0.5 | 0.00076 | 0.49 |
$10^{2}\cdot\delta t$ | CPU time | $N_{L^2}(u)$ | rate | $N_{L^2}(H)$ | rate | $N_{H^1}(u)$ | rate | $N_{H^1}(H)$ | rate |
1/1 | 00:00:01 | 0.00013 | - | 0.00012 | - | 0.01226 | - | 0.011863 | - |
1/4 | 00:00:24 | 3.4e-05 | 0.98 | 3.2e-05 | 0.98 | 0.00624 | 0.49 | 0.00603 | 0.49 |
1/16 | 00:07:44 | 9e-06 | 0.99 | 8e-06 | 0.99 | 0.00314 | 0.49 | 0.00303 | 0.49 |
1/64 | 01:29:12 | 2e-06 | 0.99 | 2e-06 | 0.99 | 0.00157 | 0.49 | 0.00152 | 0.49 |
1/256 | 23:22:36 | 1e-06 | 1 | 1e-06 | 0.99 | 0.00079 | 0.5 | 0.00076 | 0.49 |
$10^{2}\cdot\delta t$ | CPU time | $N_{L^2}(u)$ | rate | $N_{L^2}(H)$ | rate | $N_{H^1}(u)$ | rate | $N_{H^1}(H)$ | rate |
1/1 | 00:00:01 | 0.00013 | - | 0.00012 | - | 0.01226 | - | 0.011863 | - |
1/4 | 00:00:24 | 3.4e-05 | 0.98 | 3.2e-05 | 0.98 | 0.00624 | 0.49 | 0.00603 | 0.49 |
1/16 | 00:07:44 | 9e-06 | 0.99 | 8e-06 | 0.99 | 0.00314 | 0.49 | 0.00303 | 0.49 |
1/64 | 01:29:12 | 2e-06 | 0.99 | 2e-06 | 0.99 | 0.00157 | 0.49 | 0.00152 | 0.49 |
1/256 | 23:22:36 | 1e-06 | 1 | 1e-06 | 0.99 | 0.00079 | 0.5 | 0.00076 | 0.49 |
$f_H$ | $-35$ | $-.1$ | $0$ | $.1$ | $.2$ | $1.5$ | $15$ | $35$ |
log | explose | $-.41$ | $[-.37, .37]$ | $.41$ | .44 | .62 | $.94$ | explose |
pol | explose | $-.40$ | $[-.37, .37]$ | $.40$ | .43 | .64 | explose | explose |
$f_H$ | $-35$ | $-.1$ | $0$ | $.1$ | $.2$ | $1.5$ | $15$ | $35$ |
log | explose | $-.41$ | $[-.37, .37]$ | $.41$ | .44 | .62 | $.94$ | explose |
pol | explose | $-.40$ | $[-.37, .37]$ | $.40$ | .43 | .64 | explose | explose |
$f_H$ | $-35$ | $-.1$ | $0$ | $.1$ | $.2$ | $1.5$ | $15$ | $35$ |
log | $-.76$ | $[-.49, .49]$ | $[-.49, .49]$ | $[-.49, .49]$ | .49 | .52 | $.67$ | $.76$ |
pol | $-.86$ | $[-.49, .49]$ | $[-.49, .49]$ | $[-.49, .49]$ | $.49$ | $.52$ | $.70$ | $.86$ |
$f_H$ | $-35$ | $-.1$ | $0$ | $.1$ | $.2$ | $1.5$ | $15$ | $35$ |
log | $-.76$ | $[-.49, .49]$ | $[-.49, .49]$ | $[-.49, .49]$ | .49 | .52 | $.67$ | $.76$ |
pol | $-.86$ | $[-.49, .49]$ | $[-.49, .49]$ | $[-.49, .49]$ | $.49$ | $.52$ | $.70$ | $.86$ |
$f_H$ | $-35$ | $-.1$ | $0$ | $.1$ | $.2$ | $1.5$ | $15$ | $35$ |
log | $-.56$ | $[-.50, .50]$ | $[-.50, .50]$ | $[-.50, .50]$ | $[-.50, .50]$ | $[-.50, .50]$ | $.53$ | $.56$ |
pol | $-.57$ | $[-.50, .50]$ | $[-.50, .50]$ | $[-.50, .50]$ | $[-.50, .50]$ | $[-.50, .50]$ | $.53$ | $.57$ |
$f_H$ | $-35$ | $-.1$ | $0$ | $.1$ | $.2$ | $1.5$ | $15$ | $35$ |
log | $-.56$ | $[-.50, .50]$ | $[-.50, .50]$ | $[-.50, .50]$ | $[-.50, .50]$ | $[-.50, .50]$ | $.53$ | $.56$ |
pol | $-.57$ | $[-.50, .50]$ | $[-.50, .50]$ | $[-.50, .50]$ | $[-.50, .50]$ | $[-.50, .50]$ | $.53$ | $.57$ |
$f_H$ | $-35$ | $-.1$ | $0$ | $.1$ | $.2$ | $1.5$ | $15$ | $35$ |
log | explose | explose | explose | explose | .92 | .94 | .99 | explose |
pol | explose | explose | explose | explose | explose | .94 | explose | explose |
$f_H$ | $-35$ | $-.1$ | $0$ | $.1$ | $.2$ | $1.5$ | $15$ | $35$ |
log | explose | explose | explose | explose | .92 | .94 | .99 | explose |
pol | explose | explose | explose | explose | explose | .94 | explose | explose |
$f_H$ | $-35$ | $-.1$ | $0$ | $.1$ | $.2$ | $1.5$ | $15$ | $35$ |
log | -.95 | $[-.93, .93]$ | $[-.93, .93]$ | $[-.93, .93]$ | $[-.93, .93]$ | explose | .94 | .95 |
pol | explose | explose | explose | explose | explose | explose | .96 | explose |
$f_H$ | $-35$ | $-.1$ | $0$ | $.1$ | $.2$ | $1.5$ | $15$ | $35$ |
log | -.95 | $[-.93, .93]$ | $[-.93, .93]$ | $[-.93, .93]$ | $[-.93, .93]$ | explose | .94 | .95 |
pol | explose | explose | explose | explose | explose | explose | .96 | explose |
$f_H$ | $-35$ | $-.1$ | $0$ | $.1$ | $.2$ | $1.5$ | $15$ | $35$ |
log | -.93 | $[-.93, .93]$ | $[-.93, .93]$ | $[-.93, .93]$ | $[-.93, .93]$ | $[-.93, .93]$ | explose | .93 |
pol | -.94 | explose | explose | explose | explose | explose | explose | .94 |
$f_H$ | $-35$ | $-.1$ | $0$ | $.1$ | $.2$ | $1.5$ | $15$ | $35$ |
log | -.93 | $[-.93, .93]$ | $[-.93, .93]$ | $[-.93, .93]$ | $[-.93, .93]$ | $[-.93, .93]$ | explose | .93 |
pol | -.94 | explose | explose | explose | explose | explose | explose | .94 |
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