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Memory effects in measure transport equations
Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation
Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden |
In this paper we present a numerical comparison of various mass-conservative discretizations for the time-dependent Gross-Pitaevskii equation. We have three main objectives. First, we want to clarify how purely mass-conservative methods perform compared to methods that are additionally energy-conservative or symplectic. Second, we shall compare the accuracy of energy-conservative and symplectic methods among each other. Third, we will investigate if a linearized energy-conserving method suffers from a loss of accuracy compared to an approach which requires to solve a full nonlinear problem in each time-step. In order to obtain a representative comparison, our numerical experiments cover different physically relevant test cases, such as traveling solitons, stationary multi-solitons, Bose-Einstein condensates in an optical lattice and vortex pattern in a rapidly rotating superfluid. We shall also consider a computationally severe test case involving a pseudo Mott insulator. Our space discretization is based on finite elements throughout the paper. We will also give special attention to long time behavior and possible coupling conditions between time-step sizes and mesh sizes. The main observation of this paper is that mass conservation alone will not lead to a competitive method in complex settings. Furthermore, energy-conserving and symplectic methods are both reliable and accurate, yet, the energy-conservative schemes achieve a visibly higher accuracy in our test cases. Finally, the scheme that performs best throughout our experiments is an energy-conserving relaxation scheme with linear time-stepping proposed by C. Besse (SINUM, 42(3):934–952, 2004).
References:
[1] |
A. Aftalion, Vortices in Bose-Einstein Condensates, Progress in Nonlinear Differential Equations and their Applications, 67. Birkhäuser Boston, Inc., Boston, MA, 2006. |
[2] |
P. L. Christiansen, M. P. Sorensen and A. C. Scott, Nonlinear Science at the Dawn of the 21st Century, Lecture Notes in Physics, 542. Springer-Verlag, Berlin, 2000.
doi: 10.1007/3-540-46629-0. |
[3] |
G. D. Akrivis, V. A. Dougalis and O. A. Karakashian,
On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31-53.
doi: 10.1007/BF01385769. |
[4] |
T. Aktosun, T. Busse, F. Demontis and C. van der Mee, Exact solutions to the nonlinear Schrödinger equation, in Topics in Operator Theory, Systems and Mathematical Physics, Oper. Theory Adv. Appl., Birkhäuser Verlag, Basel, 2 (2010), 1–12.
doi: 10.1007/978-3-0346-0161-0_1. |
[5] |
R. Altmann, P. Henning and D. Peterseim, Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials, submitted, 2018, arXiv: 1803.09950. |
[6] |
P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev., 109 (1958), 1492–1505, URL https://link.aps.org/doi/10.1103/PhysRev.109.1492.
doi: 10.1142/9789812567154_0007. |
[7] |
X. Antoine, W. Z. Bao and C. Besse,
Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184 (2013), 2621-2633.
doi: 10.1016/j.cpc.2013.07.012. |
[8] |
D. N. Arnold, G. David, D. Jerison, S. Mayboroda and M. Filoche,
Effective confining potential of quantum states in disordered media, Phys. Rev. Lett., 116 (2016), 056602.
doi: 10.1103/PhysRevLett.116.056602. |
[9] |
W. Z. Bao and Y. Y. Cai,
Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.
doi: 10.3934/krm.2013.6.1. |
[10] |
W. Z. Bao and Q. Du,
Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697.
doi: 10.1137/S1064827503422956. |
[11] |
W. Z. Bao, S. Jin and P. A. Markowich,
Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), 27-64.
doi: 10.1137/S1064827501393253. |
[12] |
C. Besse,
A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 42 (2004), 934-952.
doi: 10.1137/S0036142901396521. |
[13] |
C. Besse, S. Descombes, G. Dujardin and I. Lacroix-Violet, Energy Preserving Methods for Nonlinear Schrödinger Equations, 2018, arXiv: 1812.04890. |
[14] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[15] |
Q. S. Chang, E. Jia and W. Sun,
Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148 (1999), 397-415.
doi: 10.1006/jcph.1998.6120. |
[16] |
D. Clément, A. F. Varón, J. A. Retter, L. Sanchez-Palencia, A Aspect and P. Bouyer, Experimental study of the transport of coherent interacting matter-waves in a 1d random potential induced by laser speckle, New Journal of Physics, 8 (2006), 165, URL http://stacks.iop.org/1367-2630/8/i=8/a=165. |
[17] |
L. Erdös, B. Schlein and H.-T. Yau,
Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math., 172 (2010), 291-370.
doi: 10.4007/annals.2010.172.291. |
[18] |
D. L. Feder, A. A. Svidzinsky, A. L. Fetter and C. W. Clark,
Anomalous modes drive vortex dynamics in confined Bose-Einstein condensates, Phys. Rev. Lett., 86 (2001), 564-567.
doi: 10.1103/PhysRevLett.86.564. |
[19] |
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature, 415 (2002), 39–44, URL http://dx.doi.org/10.1038/415039a. |
[20] |
D. F. Griffiths, A. R. Mitchell and J. L. Morris,
A numerical study of the nonlinear Schrödinger equation, Comput. Methods Appl. Mech. Engrg., 45 (1984), 177-215.
doi: 10.1016/0045-7825(84)90156-7. |
[21] |
H. Hasimoto and H. Ono,
Nonlinear modulation of gravity waves, Journal of the Physical Society of Japan, 33 (1972), 805-811.
doi: 10.1143/JPSJ.33.805. |
[22] |
P. Henning and A. Målqvist,
The finite element method for the time-dependent Gross-Pitaevskii equation with angular momentum rotation, SIAM J. Numer. Anal., 55 (2017), 923-952.
doi: 10.1137/15M1009172. |
[23] |
P. Henning and D. Peterseim,
Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials, Math. Models Methods Appl. Sci., 27 (2017), 2147-2184.
doi: 10.1142/S0218202517500415. |
[24] |
E. Jarlebring, S. Kvaal and W. Michiels,
An inverse iteration method for eigenvalue problems with eigenvector nonlinearities, SIAM J. Sci. Comput., 36 (2014), A1978-A2001.
doi: 10.1137/130910014. |
[25] |
O. Karakashian and C. Makridakis,
A space-time finite element method for the nonlinear Schrödinger equation: The continuous Galerkin method, SIAM J. Numer. Anal., 36 (1999), 1779-1807.
doi: 10.1137/S0036142997330111. |
[26] |
C. A. Mülle and D. Delande, Disorder and interference: Localization phenomena, arXiv e-prints. |
[27] |
J. M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schrödinger equation, Math. Comp., 43 (1984), 21–27, URL http://dx.doi.org/10.2307/2007397.
doi: 10.1090/S0025-5718-1984-0744922-X. |
[28] |
J. M. Sanz-Serna,
Runge-Kutta schemes for Hamiltonian systems, BIT, 28 (1988), 877-883.
doi: 10.1007/BF01954907. |
[29] |
J. M. Sanz-Serna and J. G. Verwer,
Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. Numer. Anal., 6 (1986), 25-42.
doi: 10.1093/imanum/6.1.25. |
[30] |
Y. Tourigny,
Optimal H1 estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation, IMA J. Numer. Anal., 11 (1991), 509-523.
doi: 10.1093/imanum/11.4.509. |
[31] |
J. G. Verwer and J. M. Sanz-Serna,
Convergence of method of lines approximations to partial differential equations, Computing, 33 (1984), 297-313.
doi: 10.1007/BF02242274. |
[32] |
J. Wang,
A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Sci. Comput., 60 (2014), 390-407.
doi: 10.1007/s10915-013-9799-4. |
[33] |
H. C. Yuen and B. M. Lake,
Instabilities of waves on deep water, Annual Review of Fluid Mechanics, 12 (1980), 303-334.
doi: 10.1146/annurev.fl.12.010180.001511. |
[34] |
V. E. Zakharov,
Stability of periodic waves of finite amplitude on a surface of deep fluid, Journal of Applied Mechanics and Technical Physics, 9 (1968), 190-194.
doi: 10.1007/BF00913182. |
[35] |
V. E. Zakharov and A. B. Shabat,
Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Éksper. Teoret. Fiz., 61 (1971), 118-134.
|
[36] |
G. Zhong and J. E. Marsden,
Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Physics Letters A, 133 (1988), 134-139.
doi: 10.1016/0375-9601(88)90773-6. |
[37] |
G. E. Zouraris,
On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation, M2AN Math. Model. Numer. Anal., 35 (2001), 389-405.
doi: 10.1051/m2an:2001121. |
[38] |
G. E. Zouraris, Error Estimation of the Besse Relaxation Scheme for a Semilinear Heat Equation, 2018, arXiv: 1812.09273. |
show all references
References:
[1] |
A. Aftalion, Vortices in Bose-Einstein Condensates, Progress in Nonlinear Differential Equations and their Applications, 67. Birkhäuser Boston, Inc., Boston, MA, 2006. |
[2] |
P. L. Christiansen, M. P. Sorensen and A. C. Scott, Nonlinear Science at the Dawn of the 21st Century, Lecture Notes in Physics, 542. Springer-Verlag, Berlin, 2000.
doi: 10.1007/3-540-46629-0. |
[3] |
G. D. Akrivis, V. A. Dougalis and O. A. Karakashian,
On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31-53.
doi: 10.1007/BF01385769. |
[4] |
T. Aktosun, T. Busse, F. Demontis and C. van der Mee, Exact solutions to the nonlinear Schrödinger equation, in Topics in Operator Theory, Systems and Mathematical Physics, Oper. Theory Adv. Appl., Birkhäuser Verlag, Basel, 2 (2010), 1–12.
doi: 10.1007/978-3-0346-0161-0_1. |
[5] |
R. Altmann, P. Henning and D. Peterseim, Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials, submitted, 2018, arXiv: 1803.09950. |
[6] |
P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev., 109 (1958), 1492–1505, URL https://link.aps.org/doi/10.1103/PhysRev.109.1492.
doi: 10.1142/9789812567154_0007. |
[7] |
X. Antoine, W. Z. Bao and C. Besse,
Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184 (2013), 2621-2633.
doi: 10.1016/j.cpc.2013.07.012. |
[8] |
D. N. Arnold, G. David, D. Jerison, S. Mayboroda and M. Filoche,
Effective confining potential of quantum states in disordered media, Phys. Rev. Lett., 116 (2016), 056602.
doi: 10.1103/PhysRevLett.116.056602. |
[9] |
W. Z. Bao and Y. Y. Cai,
Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.
doi: 10.3934/krm.2013.6.1. |
[10] |
W. Z. Bao and Q. Du,
Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697.
doi: 10.1137/S1064827503422956. |
[11] |
W. Z. Bao, S. Jin and P. A. Markowich,
Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), 27-64.
doi: 10.1137/S1064827501393253. |
[12] |
C. Besse,
A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 42 (2004), 934-952.
doi: 10.1137/S0036142901396521. |
[13] |
C. Besse, S. Descombes, G. Dujardin and I. Lacroix-Violet, Energy Preserving Methods for Nonlinear Schrödinger Equations, 2018, arXiv: 1812.04890. |
[14] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[15] |
Q. S. Chang, E. Jia and W. Sun,
Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148 (1999), 397-415.
doi: 10.1006/jcph.1998.6120. |
[16] |
D. Clément, A. F. Varón, J. A. Retter, L. Sanchez-Palencia, A Aspect and P. Bouyer, Experimental study of the transport of coherent interacting matter-waves in a 1d random potential induced by laser speckle, New Journal of Physics, 8 (2006), 165, URL http://stacks.iop.org/1367-2630/8/i=8/a=165. |
[17] |
L. Erdös, B. Schlein and H.-T. Yau,
Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math., 172 (2010), 291-370.
doi: 10.4007/annals.2010.172.291. |
[18] |
D. L. Feder, A. A. Svidzinsky, A. L. Fetter and C. W. Clark,
Anomalous modes drive vortex dynamics in confined Bose-Einstein condensates, Phys. Rev. Lett., 86 (2001), 564-567.
doi: 10.1103/PhysRevLett.86.564. |
[19] |
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature, 415 (2002), 39–44, URL http://dx.doi.org/10.1038/415039a. |
[20] |
D. F. Griffiths, A. R. Mitchell and J. L. Morris,
A numerical study of the nonlinear Schrödinger equation, Comput. Methods Appl. Mech. Engrg., 45 (1984), 177-215.
doi: 10.1016/0045-7825(84)90156-7. |
[21] |
H. Hasimoto and H. Ono,
Nonlinear modulation of gravity waves, Journal of the Physical Society of Japan, 33 (1972), 805-811.
doi: 10.1143/JPSJ.33.805. |
[22] |
P. Henning and A. Målqvist,
The finite element method for the time-dependent Gross-Pitaevskii equation with angular momentum rotation, SIAM J. Numer. Anal., 55 (2017), 923-952.
doi: 10.1137/15M1009172. |
[23] |
P. Henning and D. Peterseim,
Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials, Math. Models Methods Appl. Sci., 27 (2017), 2147-2184.
doi: 10.1142/S0218202517500415. |
[24] |
E. Jarlebring, S. Kvaal and W. Michiels,
An inverse iteration method for eigenvalue problems with eigenvector nonlinearities, SIAM J. Sci. Comput., 36 (2014), A1978-A2001.
doi: 10.1137/130910014. |
[25] |
O. Karakashian and C. Makridakis,
A space-time finite element method for the nonlinear Schrödinger equation: The continuous Galerkin method, SIAM J. Numer. Anal., 36 (1999), 1779-1807.
doi: 10.1137/S0036142997330111. |
[26] |
C. A. Mülle and D. Delande, Disorder and interference: Localization phenomena, arXiv e-prints. |
[27] |
J. M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schrödinger equation, Math. Comp., 43 (1984), 21–27, URL http://dx.doi.org/10.2307/2007397.
doi: 10.1090/S0025-5718-1984-0744922-X. |
[28] |
J. M. Sanz-Serna,
Runge-Kutta schemes for Hamiltonian systems, BIT, 28 (1988), 877-883.
doi: 10.1007/BF01954907. |
[29] |
J. M. Sanz-Serna and J. G. Verwer,
Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. Numer. Anal., 6 (1986), 25-42.
doi: 10.1093/imanum/6.1.25. |
[30] |
Y. Tourigny,
Optimal H1 estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation, IMA J. Numer. Anal., 11 (1991), 509-523.
doi: 10.1093/imanum/11.4.509. |
[31] |
J. G. Verwer and J. M. Sanz-Serna,
Convergence of method of lines approximations to partial differential equations, Computing, 33 (1984), 297-313.
doi: 10.1007/BF02242274. |
[32] |
J. Wang,
A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Sci. Comput., 60 (2014), 390-407.
doi: 10.1007/s10915-013-9799-4. |
[33] |
H. C. Yuen and B. M. Lake,
Instabilities of waves on deep water, Annual Review of Fluid Mechanics, 12 (1980), 303-334.
doi: 10.1146/annurev.fl.12.010180.001511. |
[34] |
V. E. Zakharov,
Stability of periodic waves of finite amplitude on a surface of deep fluid, Journal of Applied Mechanics and Technical Physics, 9 (1968), 190-194.
doi: 10.1007/BF00913182. |
[35] |
V. E. Zakharov and A. B. Shabat,
Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Éksper. Teoret. Fiz., 61 (1971), 118-134.
|
[36] |
G. Zhong and J. E. Marsden,
Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Physics Letters A, 133 (1988), 134-139.
doi: 10.1016/0375-9601(88)90773-6. |
[37] |
G. E. Zouraris,
On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation, M2AN Math. Model. Numer. Anal., 35 (2001), 389-405.
doi: 10.1051/m2an:2001121. |
[38] |
G. E. Zouraris, Error Estimation of the Besse Relaxation Scheme for a Semilinear Heat Equation, 2018, arXiv: 1812.09273. |










![]() |
IM-FEM | CN-FEM | RE-FEM | LCN-FEM | Two-Step FEM |
Mass-Conservative | Yes | Yes | Yes | Yes | Yes |
Energy-Conservative | No | Yes | Yes | No | No |
Symplectic | Yes | No | No | No | No |
Linear time-steps | No | No | Yes | Yes | Yes |
Optimal | Optimal | - | Optimal | Optimal | |
Conditional | Unconditional | Unconditional | Conditional | ||
Optimal | - | - | Optimal | Optimal | |
Conditional | Unconditional | Conditional |
![]() |
IM-FEM | CN-FEM | RE-FEM | LCN-FEM | Two-Step FEM |
Mass-Conservative | Yes | Yes | Yes | Yes | Yes |
Energy-Conservative | No | Yes | Yes | No | No |
Symplectic | Yes | No | No | No | No |
Linear time-steps | No | No | Yes | Yes | Yes |
Optimal | Optimal | - | Optimal | Optimal | |
Conditional | Unconditional | Unconditional | Conditional | ||
Optimal | - | - | Optimal | Optimal | |
Conditional | Unconditional | Conditional |
CPU [s] times 1D | |||||
IM-FEM | CN-FEM | RE-FEM | LCN-FEM | TwoStep FEM | |
1600 | 47.92 | 46.58 | 5.78 | 5.40 | 5.50 |
3200 | 96.59 | 92.57 | 11.52 | 10.75 | 10.91 |
6400 | 197.89 | 192.53 | 23.38 | 21.83 | 22.18 |
CPU [s] times 1D | |||||
IM-FEM | CN-FEM | RE-FEM | LCN-FEM | TwoStep FEM | |
1600 | 47.92 | 46.58 | 5.78 | 5.40 | 5.50 |
3200 | 96.59 | 92.57 | 11.52 | 10.75 | 10.91 |
6400 | 197.89 | 192.53 | 23.38 | 21.83 | 22.18 |
IM-FEM | CN-FEM | RE-FEM | LCN-FEM | TwoStepFEM | SP2 | ||
2.684e14 | 3.368e12 | 12.068 | 21.033 | 11.757 | 11.582 | ||
23.911 | 22.546 | 13.774 | 18.639 | 11.0617 | 6.2246 | ||
23.325 | 19.659 | 7.544 | 18.145 | 19.100 | 2.1756 | ||
15.885 | 12.738 | 4.648 | 18.525 | 15.223 | 0.5953 | ||
8.498 | 5.355 | 1.265 | 4.868 | 7.733 | 0.1523 | ||
3.356 | 1.442 | 0.319 | 8.109 | 2.301 | 0.0383 | ||
0.959 | 0.360 | 0.080 | 3.537 | 0.577 | 0.0096 | ||
0.252 | 0.091 | 0.021 | 0.723 | 0.145 | 0.0024 |
IM-FEM | CN-FEM | RE-FEM | LCN-FEM | TwoStepFEM | SP2 | ||
2.684e14 | 3.368e12 | 12.068 | 21.033 | 11.757 | 11.582 | ||
23.911 | 22.546 | 13.774 | 18.639 | 11.0617 | 6.2246 | ||
23.325 | 19.659 | 7.544 | 18.145 | 19.100 | 2.1756 | ||
15.885 | 12.738 | 4.648 | 18.525 | 15.223 | 0.5953 | ||
8.498 | 5.355 | 1.265 | 4.868 | 7.733 | 0.1523 | ||
3.356 | 1.442 | 0.319 | 8.109 | 2.301 | 0.0383 | ||
0.959 | 0.360 | 0.080 | 3.537 | 0.577 | 0.0096 | ||
0.252 | 0.091 | 0.021 | 0.723 | 0.145 | 0.0024 |
IM-FEM | CN-FEM | RE-FEM | LCN-FEM | TwoStepFEM | SP2 | ||
3.795e10 | 4.270e9 | 11.445 | 24.614 | 10.629 | 9.8755 | ||
10.368 | 11.108 | 15.194 | 22.414 | 11.157 | 4.6047 | ||
9.544 | 11.474 | 13.620 | 29.591 | 10.877 | 1.7293 | ||
10.400 | 9.935 | 3.768 | 36.725 | 10.542 | 0.4877 | ||
6.175 | 4.662 | 1.022 | 9.990 | 6.601 | 0.1258 | ||
2.704 | 1.261 | 0.260 | 12.261 | 2.043 | 0.0317 | ||
0.806 | 0.315 | 0.066 | 3.277 | 0.514 | 0.0079 | ||
0.214 | 0.080 | 0.018 | 0.632 | 0.129 | 0.0020 |
IM-FEM | CN-FEM | RE-FEM | LCN-FEM | TwoStepFEM | SP2 | ||
3.795e10 | 4.270e9 | 11.445 | 24.614 | 10.629 | 9.8755 | ||
10.368 | 11.108 | 15.194 | 22.414 | 11.157 | 4.6047 | ||
9.544 | 11.474 | 13.620 | 29.591 | 10.877 | 1.7293 | ||
10.400 | 9.935 | 3.768 | 36.725 | 10.542 | 0.4877 | ||
6.175 | 4.662 | 1.022 | 9.990 | 6.601 | 0.1258 | ||
2.704 | 1.261 | 0.260 | 12.261 | 2.043 | 0.0317 | ||
0.806 | 0.315 | 0.066 | 3.277 | 0.514 | 0.0079 | ||
0.214 | 0.080 | 0.018 | 0.632 | 0.129 | 0.0020 |
IM-FEM | CN-FEM | RE-FEM | LCN-FEM | Two-Step FEM | ||||||
2.401 | 1220 | 3.045 | 1130 | 3.045 | 130 | 95.175 | 130 | 9.338 | 140 | |
2.474 | 1960 | 0.980 | 1800 | 0.979 | 210 | 38.324 | 210 | 9.148 | 240 | |
0.710 | 3920 | 0.249 | 3540 | 0.249 | 370 | 1.948 | 370 | 8.942 | 450 | |
0.179 | 7800 | 0.062 | 7270 | 0.062 | 700 | 0.295 | 680 | 8.899 | 870 | |
0.044 | 10750 | 0.015 | 9800 | 0.015 | 1240 | 0.080 | 1190 | 8.683 | 1610 |
IM-FEM | CN-FEM | RE-FEM | LCN-FEM | Two-Step FEM | ||||||
2.401 | 1220 | 3.045 | 1130 | 3.045 | 130 | 95.175 | 130 | 9.338 | 140 | |
2.474 | 1960 | 0.980 | 1800 | 0.979 | 210 | 38.324 | 210 | 9.148 | 240 | |
0.710 | 3920 | 0.249 | 3540 | 0.249 | 370 | 1.948 | 370 | 8.942 | 450 | |
0.179 | 7800 | 0.062 | 7270 | 0.062 | 700 | 0.295 | 680 | 8.899 | 870 | |
0.044 | 10750 | 0.015 | 9800 | 0.015 | 1240 | 0.080 | 1190 | 8.683 | 1610 |
IM-FEM | CN-FEM | RE-FEM | LCN-FEM | |||||||||
64 | 0.6590 | 0.3560 | 1.9949 | 1.7527 | 0.3594 | 3.9750 | 1.7272 | 0.3553 | 3.9466 | 1.4095 | 1.4014 | 6.6062 |
128 | 1.1946 | 0.3496 | 2.1666 | 1.0690 | 0.3579 | 2.8822 | 1.0779 | 0.3579 | 2.9085 | 1.4500 | 1.2261 | 8.1410 |
256 | 1.1855 | 0.1893 | 3.1790 | 0.8746 | 0.1714 | 2.4329 | 0.8717 | 0.1707 | 2.4257 | 0.5402 | 0.1431 | 1.6532 |
512 | 0.3416 | 0.0515 | 0.9392 | 0.2362 | 0.0453 | 0.6699 | 0.2373 | 0.0456 | 0.6734 | 0.6026 | 0.4097 | 7.1779 |
1024 | 0.0847 | 0.0126 | 0.2333 | 0.0575 | 0.0110 | 0.1636 | 0.0599 | 0.0116 | 0.1706 | 0.0400 | 0.0118 | 0.2426 |
2048 | 0.0192 | 0.0027 | 0.0523 | 0.0110 | 0.0023 | 0.0348 | 0.0150 | 0.0029 | 0.0427 | 0.0111 | 0.0013 | 0.0239 |
IM-FEM | CN-FEM | RE-FEM | LCN-FEM | |||||||||
64 | 0.6590 | 0.3560 | 1.9949 | 1.7527 | 0.3594 | 3.9750 | 1.7272 | 0.3553 | 3.9466 | 1.4095 | 1.4014 | 6.6062 |
128 | 1.1946 | 0.3496 | 2.1666 | 1.0690 | 0.3579 | 2.8822 | 1.0779 | 0.3579 | 2.9085 | 1.4500 | 1.2261 | 8.1410 |
256 | 1.1855 | 0.1893 | 3.1790 | 0.8746 | 0.1714 | 2.4329 | 0.8717 | 0.1707 | 2.4257 | 0.5402 | 0.1431 | 1.6532 |
512 | 0.3416 | 0.0515 | 0.9392 | 0.2362 | 0.0453 | 0.6699 | 0.2373 | 0.0456 | 0.6734 | 0.6026 | 0.4097 | 7.1779 |
1024 | 0.0847 | 0.0126 | 0.2333 | 0.0575 | 0.0110 | 0.1636 | 0.0599 | 0.0116 | 0.1706 | 0.0400 | 0.0118 | 0.2426 |
2048 | 0.0192 | 0.0027 | 0.0523 | 0.0110 | 0.0023 | 0.0348 | 0.0150 | 0.0029 | 0.0427 | 0.0111 | 0.0013 | 0.0239 |
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