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December  2020, 7(2): 339-368. doi: 10.3934/jcd.2020014

On the development of symmetry-preserving finite element schemes for ordinary differential equations

1. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, A1C 5S7, Canada

2. 

Department of Mathematics, Monmouth University, West Long Branch, NJ, 07764, USA

* Corresponding author: James Jackaman

Received  July 2019 Published  July 2020

Fund Project: This research was supported, in part, thanks to the Canada Research Chairs, the InnovateNL LeverageR&D and NSERC Discovery grant programs

In this paper we introduce a procedure, based on the method of equivariant moving frames, for formulating continuous Galerkin finite element schemes that preserve the Lie point symmetries of initial value problems for ordinary differential equations. Our methodology applies to projectable and non-projectable symmetry group actions, to ordinary differential equations of arbitrary order, and finite element approximations of arbitrary polynomial degree. Several examples are included to illustrate various features of the symmetry-preserving process. We summarise extensive numerical experiments showing that symmetry-preserving finite element schemes may provide better long term accuracy than their non-invariant counterparts and can be implemented on larger elements.

Citation: Alex Bihlo, James Jackaman, Francis Valiquette. On the development of symmetry-preserving finite element schemes for ordinary differential equations. Journal of Computational Dynamics, 2020, 7 (2) : 339-368. doi: 10.3934/jcd.2020014
References:
[1]

M. I. BakirovaV. A. Dorodnitsyn and R. V. Kozlov, Symmetry-preserving difference schemes for some heat transfer equations, J. Phys. A, 30 (1997), 8139-8155.  doi: 10.1088/0305-4470/30/23/014.

[2]

A. Bihlo, Invariant meshless discretization schemes, J. Phys. A, 46 (2013), 12pp. doi: 10.1088/1751-8113/46/6/062001.

[3]

A. Bihlo, X. Coiteux-Roy and P. Winternitz, The Korteweg–de Vries equation and its symmetry-preserving discretization, J. Phys. A, 48 (2015), 25pp. doi: 10.1088/1751-8113/48/5/055201.

[4]

A. Bihlo and J.-C. Nave, Invariant discretization scheme using evolution-projection techniques, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), 23pp. doi: 10.3842/SIGMA.2013.052.

[5]

A. Bihlo and J.-C. Nave, Convecting reference frames and invariant numerical models, J. Comput. Phys., 272 (2014), 656-663.  doi: 10.1016/j.jcp.2014.04.042.

[6]

A. Bihlo and R. O. Popovych, Invariant discretization schemes for the shallow water equations, SIAM J. Sci. Comput., 34 (2012), B810-B839.  doi: 10.1137/120861187.

[7]

A. Bihlo and F. Valiquette, Symmetry-preserving numerical schemes, in Symmetries and Integrability of Difference Equations, CRM Ser. Math. Phys., Springer, Cham, 2017,261–324. doi: 10.1007/978-3-319-56666-5_6.

[8]

A. Bihlo and F. Valiquette, Symmetry-preserving finite element schemes: An introductory investigation, SIAM J. Sci. Comput., 41 (2019), A3300-A3325.  doi: 10.1137/18M1177524.

[9] S. Blanes and F. Casas, A Concise Introduction to Geometric Numerical Integration, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016. 
[10]

G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, Applied Mathematical Sciences, 154, Springer-Verlag, New York, 2002. doi: 10.1007/b97380.

[11]

A. BourliouxC. Cyr-Gagnon and P. Winternitz, Difference schemes with point symmetries and their numerical tests, J. Phys. A, 39 (2006), 6877-6896.  doi: 10.1088/0305-4470/39/22/006.

[12]

A. BourliouxR. Rebelo and P. Winternitz, Symmetry preserving discretization of $SL(2, \mathbb R)$ invariant equations, J. Nonlinear Math. Phys., 15 (2008), 362-372.  doi: 10.2991/jnmp.2008.15.s3.35.

[13]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.

[14]

C. Budd and V. Dorodnitsyn, Symmetry-adapted moving mesh schemes for the nonlinear Schrödinger equation. Symmetry and integrability of difference equations, J. Phys. A, 34 (2001), 10387-10400.  doi: 10.1088/0305-4470/34/48/305.

[15]

V. A. Dorodnitsyn, Transformation groups in difference spaces, J. Soviet Math., 55 (1991), 1490-1517.  doi: 10.1007/BF01097535.

[16] V. Dorodnitsyn, Applications of Lie Groups to Difference Equations, Differential and Integral Equations and Their Applications, 8, CRC Press, Boca Raton, FL, 2011. 
[17]

V. Dorodnitsyn and P. Winternitz, Lie point symmetry preserving discretization for variable coefficient Korteweg-de Vries equations. Modern group analysis, Nonlinear Dynam., 22 (2000), 49-59.  doi: 10.1023/A:1008365224018.

[18]

D. Estep, A posteriori error bounds and global error control for approximation of ordinary differential equations, SIAM J. Numer. Anal., 32 (1995), 1-48.  doi: 10.1137/0732001.

[19]

D. Estep and D. French, Global error control for the continuous Galerkin finite element method for ordinary differential equations, RAIRO Modél. Math. Anal. Numér., 28 (1994), 815-852.  doi: 10.1051/m2an/1994280708151.

[20]

D. J. Estep and A. M. Stuart, The dynamical behavior of the discontinuous Galerkin method and related difference schemes, Math. Comp., 71 (2002), 1075-1103.  doi: 10.1090/S0025-5718-01-01364-3.

[21]

M. Fels and P. J. Olver, Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math., 55 (1999), 127-208.  doi: 10.1023/A:1006195823000.

[22]

D. A. French and J. W. Schaeffer, Continuous finite element methods which preserve energy properties for nonlinear problems, Appl. Math. Comput., 39 (1990), 271-295. 

[23]

R. B. Gardner, The Method of Equivalence and its Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970135.

[24]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-30666-8.

[25]

P. Hansbo, A note on energy conservation for Hamiltonian systems using continuous time finite elements, Commun. Numer. Meth. Engrg., 17 (2001), 863-869.  doi: 10.1002/cnm.458.

[26] P. E. Hydon, Symmetry Methods for Differential Equations. A Beginner's Guide, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511623967.
[27]

J. Jackaman, Finite Element Methods as Geometric Structure Preserving Algorithms, Ph.D thesis, University of Reading, 2018.

[28]

C. Johnson, Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations, SIAM J. Numer. Anal., 25 (1988), 908-926.  doi: 10.1137/0725051.

[29]

N. Kamran, Contributions to the study of the equivalence problem of Élie Cartan and its applications to partial and ordinary differential equations, Acad. Roy. Belg. Cl. Sci. Mém. Collect. 8° (2), 45 (1989), 122pp.

[30]

P. Kim, Invariantization of the Crank-Nicolson method for Burgers' equation, Phys. D, 237 (2008), 243-254.  doi: 10.1016/j.physd.2007.09.001.

[31]

P. Kim and P. J. Olver, Geometric integration via multi-space, Regul. Chaotic Dyn., 9 (2004), 213-226.  doi: 10.1070/RD2004v009n03ABEH000277.

[32]

I. A. Kogan and P. J. Olver, Invariant Euler–Lagrange equations and the invariant variational bicomplex, Acta Appl. Math., 76 (2003), 137-193.  doi: 10.1023/A:1022993616247.

[33] B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge Monographs on Applied and Computational Mathematics, 14, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511614118.
[34]

D. Levi, L. Martina and P. Winternitz, Structure preserving discretizations of the Liouville equation and their numerical tests, SIGMA Symmetry Integrability Geom. Methods Appl., 11 (2015), 20pp. doi: 10.3842/SIGMA.2015.080.

[35] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883.
[36] E. L. Mansfield, A Practical Guide to the Invariant Calculus, Cambridge Monographs on Applied and Computational Mathematics, 26, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511844621.
[37]

G. Marí Beffa and E. L. Mansfield, Discrete moving frames on lattice varieties and lattice-based multispaces, Found. Comput. Math., 18 (2018), 181-247.  doi: 10.1007/s10208-016-9337-5.

[38]

R. I. McLachlanG. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 1021-1045.  doi: 10.1098/rsta.1999.0363.

[39]

T. E. Oliphant, A Guide to NumPy, Trelgol Publishing, USA, 2006.

[40] P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511609565.
[41]

P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.

[42]

P. J. Olver, Joint invariant signatures, Found. Comput. Math., 1 (2001), 3-67.  doi: 10.1007/s10208001001.

[43]

P. J. Olver, Invariants of finite and discrete group actions via moving frames, preprint.

[44]

P. J. Olver and J. Pohjanpelto, Moving frames for Lie pseudo-groups, Canad. J. Math., 60 (2008), 1336-1386.  doi: 10.4153/CJM-2008-057-0.

[45]

V. Ovsienko and S. Tabachnikov, What is $\ldots$ the Schwarzian derivative?, Notices Amer. Math. Soc., 56 (2009), 34-36. 

[46]

G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 7pp. doi: 10.1088/1751-8113/41/4/045206.

[47]

F. Rathgeber, D. A. Ham, L. Mitchell, M. Lange and F. Luporini, et al., Firedrake: Automating the finite element method by composing abstractions, ACM Trans. Math. Software, 43 (2017), 27pp. doi: 10.1145/2998441.

[48]

R. Rebelo and F. Valiquette, Symmetry preserving numerical schemes for partial differential equations and their numerical tests, J. Difference Equ. Appl., 19 (2103), 738-757.  doi: 10.1080/10236198.2012.685470.

[49]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Applied Mathematics and Mathematical Computation, 7, Chapman & Hall, London, 1994.

[50]

A. T. S. WanA. Bihlo and J.-C. Nave, The multiplier method to construct conservative finite difference schemes for ordinary and partial differential equations, SIAM J. Numer. Anal., 54 (2016), 86-119.  doi: 10.1137/140997944.

[51]

A. T. S. WanA. Bihlo and J.-C. Nave, Conservative methods for dynamical systems, SIAM J. Numer. Anal., 55 (2017), 2255-2285.  doi: 10.1137/16M110719X.

[52]

G. Zhong and J. E. Marsden, Lie–Poisson, Hamilton–Jacobi theory and Lie–Poisson integrators, Phys. Lett. A, 133 (1988), 134-139.  doi: 10.1016/0375-9601(88)90773-6.

[53]

B. Zhou and C.-J. Zhu, An application of the Schwarzian derivative, preprint, arXiv: hep-th/9907193.

[54]

B. Zhou and C.-J. Zhu, The complete brane solution in $D$-dimensional coupled gravity system, Comm. Theor. Phys., 32 (1999). doi: 10.1088/0253-6102/32/2/173.

show all references

References:
[1]

M. I. BakirovaV. A. Dorodnitsyn and R. V. Kozlov, Symmetry-preserving difference schemes for some heat transfer equations, J. Phys. A, 30 (1997), 8139-8155.  doi: 10.1088/0305-4470/30/23/014.

[2]

A. Bihlo, Invariant meshless discretization schemes, J. Phys. A, 46 (2013), 12pp. doi: 10.1088/1751-8113/46/6/062001.

[3]

A. Bihlo, X. Coiteux-Roy and P. Winternitz, The Korteweg–de Vries equation and its symmetry-preserving discretization, J. Phys. A, 48 (2015), 25pp. doi: 10.1088/1751-8113/48/5/055201.

[4]

A. Bihlo and J.-C. Nave, Invariant discretization scheme using evolution-projection techniques, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), 23pp. doi: 10.3842/SIGMA.2013.052.

[5]

A. Bihlo and J.-C. Nave, Convecting reference frames and invariant numerical models, J. Comput. Phys., 272 (2014), 656-663.  doi: 10.1016/j.jcp.2014.04.042.

[6]

A. Bihlo and R. O. Popovych, Invariant discretization schemes for the shallow water equations, SIAM J. Sci. Comput., 34 (2012), B810-B839.  doi: 10.1137/120861187.

[7]

A. Bihlo and F. Valiquette, Symmetry-preserving numerical schemes, in Symmetries and Integrability of Difference Equations, CRM Ser. Math. Phys., Springer, Cham, 2017,261–324. doi: 10.1007/978-3-319-56666-5_6.

[8]

A. Bihlo and F. Valiquette, Symmetry-preserving finite element schemes: An introductory investigation, SIAM J. Sci. Comput., 41 (2019), A3300-A3325.  doi: 10.1137/18M1177524.

[9] S. Blanes and F. Casas, A Concise Introduction to Geometric Numerical Integration, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016. 
[10]

G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, Applied Mathematical Sciences, 154, Springer-Verlag, New York, 2002. doi: 10.1007/b97380.

[11]

A. BourliouxC. Cyr-Gagnon and P. Winternitz, Difference schemes with point symmetries and their numerical tests, J. Phys. A, 39 (2006), 6877-6896.  doi: 10.1088/0305-4470/39/22/006.

[12]

A. BourliouxR. Rebelo and P. Winternitz, Symmetry preserving discretization of $SL(2, \mathbb R)$ invariant equations, J. Nonlinear Math. Phys., 15 (2008), 362-372.  doi: 10.2991/jnmp.2008.15.s3.35.

[13]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.

[14]

C. Budd and V. Dorodnitsyn, Symmetry-adapted moving mesh schemes for the nonlinear Schrödinger equation. Symmetry and integrability of difference equations, J. Phys. A, 34 (2001), 10387-10400.  doi: 10.1088/0305-4470/34/48/305.

[15]

V. A. Dorodnitsyn, Transformation groups in difference spaces, J. Soviet Math., 55 (1991), 1490-1517.  doi: 10.1007/BF01097535.

[16] V. Dorodnitsyn, Applications of Lie Groups to Difference Equations, Differential and Integral Equations and Their Applications, 8, CRC Press, Boca Raton, FL, 2011. 
[17]

V. Dorodnitsyn and P. Winternitz, Lie point symmetry preserving discretization for variable coefficient Korteweg-de Vries equations. Modern group analysis, Nonlinear Dynam., 22 (2000), 49-59.  doi: 10.1023/A:1008365224018.

[18]

D. Estep, A posteriori error bounds and global error control for approximation of ordinary differential equations, SIAM J. Numer. Anal., 32 (1995), 1-48.  doi: 10.1137/0732001.

[19]

D. Estep and D. French, Global error control for the continuous Galerkin finite element method for ordinary differential equations, RAIRO Modél. Math. Anal. Numér., 28 (1994), 815-852.  doi: 10.1051/m2an/1994280708151.

[20]

D. J. Estep and A. M. Stuart, The dynamical behavior of the discontinuous Galerkin method and related difference schemes, Math. Comp., 71 (2002), 1075-1103.  doi: 10.1090/S0025-5718-01-01364-3.

[21]

M. Fels and P. J. Olver, Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math., 55 (1999), 127-208.  doi: 10.1023/A:1006195823000.

[22]

D. A. French and J. W. Schaeffer, Continuous finite element methods which preserve energy properties for nonlinear problems, Appl. Math. Comput., 39 (1990), 271-295. 

[23]

R. B. Gardner, The Method of Equivalence and its Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970135.

[24]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-30666-8.

[25]

P. Hansbo, A note on energy conservation for Hamiltonian systems using continuous time finite elements, Commun. Numer. Meth. Engrg., 17 (2001), 863-869.  doi: 10.1002/cnm.458.

[26] P. E. Hydon, Symmetry Methods for Differential Equations. A Beginner's Guide, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511623967.
[27]

J. Jackaman, Finite Element Methods as Geometric Structure Preserving Algorithms, Ph.D thesis, University of Reading, 2018.

[28]

C. Johnson, Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations, SIAM J. Numer. Anal., 25 (1988), 908-926.  doi: 10.1137/0725051.

[29]

N. Kamran, Contributions to the study of the equivalence problem of Élie Cartan and its applications to partial and ordinary differential equations, Acad. Roy. Belg. Cl. Sci. Mém. Collect. 8° (2), 45 (1989), 122pp.

[30]

P. Kim, Invariantization of the Crank-Nicolson method for Burgers' equation, Phys. D, 237 (2008), 243-254.  doi: 10.1016/j.physd.2007.09.001.

[31]

P. Kim and P. J. Olver, Geometric integration via multi-space, Regul. Chaotic Dyn., 9 (2004), 213-226.  doi: 10.1070/RD2004v009n03ABEH000277.

[32]

I. A. Kogan and P. J. Olver, Invariant Euler–Lagrange equations and the invariant variational bicomplex, Acta Appl. Math., 76 (2003), 137-193.  doi: 10.1023/A:1022993616247.

[33] B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge Monographs on Applied and Computational Mathematics, 14, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511614118.
[34]

D. Levi, L. Martina and P. Winternitz, Structure preserving discretizations of the Liouville equation and their numerical tests, SIGMA Symmetry Integrability Geom. Methods Appl., 11 (2015), 20pp. doi: 10.3842/SIGMA.2015.080.

[35] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883.
[36] E. L. Mansfield, A Practical Guide to the Invariant Calculus, Cambridge Monographs on Applied and Computational Mathematics, 26, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511844621.
[37]

G. Marí Beffa and E. L. Mansfield, Discrete moving frames on lattice varieties and lattice-based multispaces, Found. Comput. Math., 18 (2018), 181-247.  doi: 10.1007/s10208-016-9337-5.

[38]

R. I. McLachlanG. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 1021-1045.  doi: 10.1098/rsta.1999.0363.

[39]

T. E. Oliphant, A Guide to NumPy, Trelgol Publishing, USA, 2006.

[40] P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511609565.
[41]

P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.

[42]

P. J. Olver, Joint invariant signatures, Found. Comput. Math., 1 (2001), 3-67.  doi: 10.1007/s10208001001.

[43]

P. J. Olver, Invariants of finite and discrete group actions via moving frames, preprint.

[44]

P. J. Olver and J. Pohjanpelto, Moving frames for Lie pseudo-groups, Canad. J. Math., 60 (2008), 1336-1386.  doi: 10.4153/CJM-2008-057-0.

[45]

V. Ovsienko and S. Tabachnikov, What is $\ldots$ the Schwarzian derivative?, Notices Amer. Math. Soc., 56 (2009), 34-36. 

[46]

G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 7pp. doi: 10.1088/1751-8113/41/4/045206.

[47]

F. Rathgeber, D. A. Ham, L. Mitchell, M. Lange and F. Luporini, et al., Firedrake: Automating the finite element method by composing abstractions, ACM Trans. Math. Software, 43 (2017), 27pp. doi: 10.1145/2998441.

[48]

R. Rebelo and F. Valiquette, Symmetry preserving numerical schemes for partial differential equations and their numerical tests, J. Difference Equ. Appl., 19 (2103), 738-757.  doi: 10.1080/10236198.2012.685470.

[49]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Applied Mathematics and Mathematical Computation, 7, Chapman & Hall, London, 1994.

[50]

A. T. S. WanA. Bihlo and J.-C. Nave, The multiplier method to construct conservative finite difference schemes for ordinary and partial differential equations, SIAM J. Numer. Anal., 54 (2016), 86-119.  doi: 10.1137/140997944.

[51]

A. T. S. WanA. Bihlo and J.-C. Nave, Conservative methods for dynamical systems, SIAM J. Numer. Anal., 55 (2017), 2255-2285.  doi: 10.1137/16M110719X.

[52]

G. Zhong and J. E. Marsden, Lie–Poisson, Hamilton–Jacobi theory and Lie–Poisson integrators, Phys. Lett. A, 133 (1988), 134-139.  doi: 10.1016/0375-9601(88)90773-6.

[53]

B. Zhou and C.-J. Zhu, An application of the Schwarzian derivative, preprint, arXiv: hep-th/9907193.

[54]

B. Zhou and C.-J. Zhu, The complete brane solution in $D$-dimensional coupled gravity system, Comm. Theor. Phys., 32 (1999). doi: 10.1088/0253-6102/32/2/173.

Figure 1.  Absolute difference between the exact solution and the standard scheme (18) and the invariant scheme (36), with $ q = 0 $ and $ {\tau}{} = 0.25 $
Figure 2.  Absolute difference between the exact solution (110) and the naive discretisation (104) and the invariant discretisation (108) with $ {\tau}{} = 0.01 $
Table 1.  The standard finite element approximation (18) where (89) and (90) hold with T = 10
q τ Maximal nodal error L2 error EOC
1.56e-01 7.49e-04 1.70e-03 -
0 7.81e-02 1.87e-04 4.25e-04 2.00
3.91e-02 4.68e-05 1.06e-04 2.00
1.95e-02 1.17e-05 2.66e-05 2.00
1.56e-01 3.04e-07 2.19e-05 -
1 7.81e-02 1.90e-08 2.74e-06 3.00
3.91e-02 1.19e-09 3.43e-07 3.00
1.95e-02 7.43e-11 4.28e-08 3.00
1.56e-01 5.31e-11 1.58e-07 -
2 7.81e-02 8.30e-13 9.91e-09 4.00
3.91e-02 1.39e-14 6.20e-10 4.00
1.95e-02 4.75e-15 3.87e-11 4.00
q τ Maximal nodal error L2 error EOC
1.56e-01 7.49e-04 1.70e-03 -
0 7.81e-02 1.87e-04 4.25e-04 2.00
3.91e-02 4.68e-05 1.06e-04 2.00
1.95e-02 1.17e-05 2.66e-05 2.00
1.56e-01 3.04e-07 2.19e-05 -
1 7.81e-02 1.90e-08 2.74e-06 3.00
3.91e-02 1.19e-09 3.43e-07 3.00
1.95e-02 7.43e-11 4.28e-08 3.00
1.56e-01 5.31e-11 1.58e-07 -
2 7.81e-02 8.30e-13 9.91e-09 4.00
3.91e-02 1.39e-14 6.20e-10 4.00
1.95e-02 4.75e-15 3.87e-11 4.00
Table 2.  The invariant finite element approximation (36) where (89) and (90) hold with T = 10
q τ Maximal nodal error L2 error EOC
1.56e-01 3.96e-16 2.23e-03 -
0 7.81e-02 2.84e-16 5.57e-04 2.00
3.91e-02 1.16e-15 1.39e-04 2.00
1.95e-02 7.77e-16 3.48e-05 2.00
1.56e-01 5.83e-16 2.19e-05 -
1 7.81e-02 5.55e-16 2.74e-06 3.00
3.91e-02 7.77e-16 3.43e-07 3.00
1.95e-02 9.99e-16 4.28e-08 3.00
1.56e-01 5.00e-16 1.58e-07 -
2 7.81e-02 1.17e-15 9.91e-09 4.00
3.91e-02 3.11e-15 6.20e-10 4.00
1.95e-02 4.77e-15 3.87e-11 4.00
q τ Maximal nodal error L2 error EOC
1.56e-01 3.96e-16 2.23e-03 -
0 7.81e-02 2.84e-16 5.57e-04 2.00
3.91e-02 1.16e-15 1.39e-04 2.00
1.95e-02 7.77e-16 3.48e-05 2.00
1.56e-01 5.83e-16 2.19e-05 -
1 7.81e-02 5.55e-16 2.74e-06 3.00
3.91e-02 7.77e-16 3.43e-07 3.00
1.95e-02 9.99e-16 4.28e-08 3.00
1.56e-01 5.00e-16 1.58e-07 -
2 7.81e-02 1.17e-15 9.91e-09 4.00
3.91e-02 3.11e-15 6.20e-10 4.00
1.95e-02 4.77e-15 3.87e-11 4.00
Table 3.  The standard finite element approximation (54) where (93), (94) and (95) hold with T = 1000
q τ Maximal nodal error L2 error EOC
1.56e-01 1.45e-01 1.27e-01 -
0 7.81e-02 7.52e-02 3.17e-02 2.00
3.91e-02 3.83e-02 7.91e-03 2.00
1.95e-02 1.93e-02 1.98e-03 2.00
1.56e-01 1.45e-01 7.79e-05 -
1 7.81e-02 7.52e-02 9.81e-06 2.99
3.91e-02 3.83e-02 1.23e-06 3.00
1.95e-02 1.93e-02 1.54e-07 3.00
1.56e-01 1.45e-01 1.48e-06 -
2 7.81e-02 7.52e-02 9.38e-08 3.98
3.91e-02 3.83e-02 5.88e-09 4.00
1.95e-02 1.93e-02 3.68e-10 4.00
q τ Maximal nodal error L2 error EOC
1.56e-01 1.45e-01 1.27e-01 -
0 7.81e-02 7.52e-02 3.17e-02 2.00
3.91e-02 3.83e-02 7.91e-03 2.00
1.95e-02 1.93e-02 1.98e-03 2.00
1.56e-01 1.45e-01 7.79e-05 -
1 7.81e-02 7.52e-02 9.81e-06 2.99
3.91e-02 3.83e-02 1.23e-06 3.00
1.95e-02 1.93e-02 1.54e-07 3.00
1.56e-01 1.45e-01 1.48e-06 -
2 7.81e-02 7.52e-02 9.38e-08 3.98
3.91e-02 3.83e-02 5.88e-09 4.00
1.95e-02 1.93e-02 3.68e-10 4.00
Table 4.  The invariant finite element approximation (59) where (93), (94) and (95) hold with T = 1000
q τ Maximal nodal error L2 error EOC
1.56e-01 1.45e-01 3.60e-03 -
0 7.81e-02 7.52e-02 9.04e-04 1.99
3.91e-02 3.83e-02 2.26e-04 2.00
1.95e-02 1.93e-02 5.66e-05 2.00
1.56e-01 1.45e-01 7.77e-05 -
1 7.81e-02 7.52e-02 9.81e-06 2.99
3.91e-02 3.83e-02 1.23e-06 3.00
1.95e-02 1.93e-02 1.54e-07 3.00
1.56e-01 1.45e-01 1.48e-06 -
2 7.81e-02 7.52e-02 9.37e-08 3.98
3.91e-02 3.83e-02 5.88e-09 4.00
1.95e-02 1.93e-02 3.79e-10 3.95
q τ Maximal nodal error L2 error EOC
1.56e-01 1.45e-01 3.60e-03 -
0 7.81e-02 7.52e-02 9.04e-04 1.99
3.91e-02 3.83e-02 2.26e-04 2.00
1.95e-02 1.93e-02 5.66e-05 2.00
1.56e-01 1.45e-01 7.77e-05 -
1 7.81e-02 7.52e-02 9.81e-06 2.99
3.91e-02 3.83e-02 1.23e-06 3.00
1.95e-02 1.93e-02 1.54e-07 3.00
1.56e-01 1.45e-01 1.48e-06 -
2 7.81e-02 7.52e-02 9.37e-08 3.98
3.91e-02 3.83e-02 5.88e-09 4.00
1.95e-02 1.93e-02 3.79e-10 3.95
Table 5.  The standard finite element approximation (65) where (96) and (97) hold
q τ Maximal nodal error L2 error EOC
1.56e-01 2.41e-01 2.48e-02 -
0 7.81e-02 1.35e-01 6.30e-03 1.98
3.91e-02 7.25e-02 1.58e-03 1.99
1.95e-02 3.76e-02 3.96e-04 2.00
1.56e-01 2.34e-01 1.22e-03 -
1 7.81e-02 1.34e-01 1.58e-04 2.94
3.91e-02 7.23e-02 2.00e-05 2.99
1.95e-02 3.75e-02 2.50e-06 3.00
1.56e-01 2.34e-01 6.22e-05 -
2 7.81e-02 1.34e-01 4.11e-06 3.92
3.91e-02 7.23e-02 2.60e-07 3.98
1.95e-02 3.75e-02 1.64e-08 3.99
q τ Maximal nodal error L2 error EOC
1.56e-01 2.41e-01 2.48e-02 -
0 7.81e-02 1.35e-01 6.30e-03 1.98
3.91e-02 7.25e-02 1.58e-03 1.99
1.95e-02 3.76e-02 3.96e-04 2.00
1.56e-01 2.34e-01 1.22e-03 -
1 7.81e-02 1.34e-01 1.58e-04 2.94
3.91e-02 7.23e-02 2.00e-05 2.99
1.95e-02 3.75e-02 2.50e-06 3.00
1.56e-01 2.34e-01 6.22e-05 -
2 7.81e-02 1.34e-01 4.11e-06 3.92
3.91e-02 7.23e-02 2.60e-07 3.98
1.95e-02 3.75e-02 1.64e-08 3.99
Table 6.  The invariant finite element approximation (70) where (96) and (97) hold
q τ Maximal nodal error L2 error EOC
1.56e-01 2.43e-01 2.33e-02 -
0 7.81e-02 1.36e-01 6.09e-03 1.94
3.91e-02 7.25e-02 1.54e-03 1.98
1.95e-02 3.76e-02 3.87e-04 2.00
1.56e-01 2.34e-01 1.26e-03 -
1 7.81e-02 1.34e-01 1.59e-04 2.99
3.91e-02 7.23e-02 2.00e-05 2.99
1.95e-02 3.75e-02 2.50e-06 3.00
1.56e-01 2.34e-01 6.24e-05 -
2 7.81e-02 1.34e-01 4.10e-06 3.93
3.91e-02 7.23e-02 2.60e-07 3.98
1.95e-02 3.75e-02 1.67e-08 3.96
q τ Maximal nodal error L2 error EOC
1.56e-01 2.43e-01 2.33e-02 -
0 7.81e-02 1.36e-01 6.09e-03 1.94
3.91e-02 7.25e-02 1.54e-03 1.98
1.95e-02 3.76e-02 3.87e-04 2.00
1.56e-01 2.34e-01 1.26e-03 -
1 7.81e-02 1.34e-01 1.59e-04 2.99
3.91e-02 7.23e-02 2.00e-05 2.99
1.95e-02 3.75e-02 2.50e-06 3.00
1.56e-01 2.34e-01 6.24e-05 -
2 7.81e-02 1.34e-01 4.10e-06 3.93
3.91e-02 7.23e-02 2.60e-07 3.98
1.95e-02 3.75e-02 1.67e-08 3.96
Table 7.  A table confirming whether the standard finite element approximation (73) and the invariant approximation (76) may be successfully solved for various step sizes τ when approximating the exact solution (99) with C = 1, y0 = 0.5
τ Standard scheme Invariant scheme
0.390625
0.78125
1.5625 ×
3.125 ×
6.25 × ×
τ Standard scheme Invariant scheme
0.390625
0.78125
1.5625 ×
3.125 ×
6.25 × ×
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