zbMATH — the first resource for mathematics

On the development of symmetry-preserving finite element schemes for ordinary differential equations. (English) Zbl 1452.65136
Summary: 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.
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
58D19 Group actions and symmetry properties
Firedrake; NumPy
Full Text: DOI
[1] M. I. Bakirova; V. A. Dorodnitsyn; R. V. Kozlov, Symmetry-preserving difference schemes for some heat transfer equations, J. Phys. A, 30, 8139-8155 (1997) · Zbl 0927.65105
[2] A. Bihlo, Invariant meshless discretization schemes, J. Phys. A, 46 (2013), 12pp. · Zbl 1263.65079
[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. · Zbl 1319.35214
[4] A. Bihlo and J.-C. Nave, Invariant discretization scheme using evolution-projection techniques, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), 23pp. · Zbl 1288.65118
[5] A. Bihlo; J.-C. Nave, Convecting reference frames and invariant numerical models, J. Comput. Phys., 272, 656-663 (2014) · Zbl 1349.76435
[6] A. Bihlo; R. O. Popovych, Invariant discretization schemes for the shallow water equations, SIAM J. Sci. Comput., 34, B810-B839 (2012) · Zbl 1263.76056
[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. · Zbl 1373.65053
[8] A. Bihlo; F. Valiquette, Symmetry-preserving finite element schemes: An introductory investigation, SIAM J. Sci. Comput., 41, A3300-A3325 (2019) · Zbl 1436.65092
[9] G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, Applied Mathematical Sciences, 154, Springer-Verlag, New York, 2002. · Zbl 1013.34004
[10] A. Bourlioux; C. Cyr-Gagnon; P. Winternitz, Difference schemes with point symmetries and their numerical tests, J. Phys. A, 39, 6877-6896 (2006) · Zbl 1095.65070
[11] A. Bourlioux; R. Rebelo; P. Winternitz, Symmetry preserving discretization of \(SL(2, \mathbb R)\) invariant equations, J. Nonlinear Math. Phys., 15, 362-372 (2008) · Zbl 1362.65076
[12] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15, Springer, New York, 2008. · Zbl 1135.65042
[13] C. Budd; V. Dorodnitsyn, Symmetry-adapted moving mesh schemes for the nonlinear Schrödinger equation. Symmetry and integrability of difference equations, J. Phys. A, 34, 10387-10400 (2001) · Zbl 0991.65085
[14] V. A. Dorodnitsyn, Transformation groups in difference spaces, J. Soviet Math., 55, 1490-1517 (1991) · Zbl 0727.65074
[15] V. Dorodnitsyn; P. Winternitz, Lie point symmetry preserving discretization for variable coefficient Korteweg-de Vries equations. Modern group analysis, Nonlinear Dynam., 22, 49-59 (2000) · Zbl 0956.65081
[16] D. Estep, A posteriori error bounds and global error control for approximation of ordinary differential equations, SIAM J. Numer. Anal., 32, 1-48 (1995) · Zbl 0820.65052
[17] D. Estep; D. French, Global error control for the continuous Galerkin finite element method for ordinary differential equations, RAIRO Modél. Math. Anal. Numér., 28, 815-852 (1994) · Zbl 0822.65054
[18] D. J. Estep; A. M. Stuart, The dynamical behavior of the discontinuous Galerkin method and related difference schemes, Math. Comp., 71, 1075-1103 (2002) · Zbl 0998.65080
[19] M. Fels; P. J. Olver, Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math., 55, 127-208 (1999) · Zbl 0937.53013
[20] D. A. French; J. W. Schaeffer, Continuous finite element methods which preserve energy properties for nonlinear problems, Appl. Math. Comput., 39, 271-295 (1990) · Zbl 0716.65084
[21] 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. · Zbl 0694.53027
[22] 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. · Zbl 1094.65125
[23] P. Hansbo, A note on energy conservation for Hamiltonian systems using continuous time finite elements, Commun. Numer. Meth. Engrg., 17, 863-869 (2001) · Zbl 0994.65136
[24] J. Jackaman, Finite Element Methods as Geometric Structure Preserving Algorithms, Ph.D thesis, University of Reading, 2018.
[25] 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, 908-926 (1988) · Zbl 0661.65076
[26] 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. · Zbl 0721.58001
[27] P. Kim, Invariantization of the Crank-Nicolson method for Burgers’ equation, Phys. D, 237, 243-254 (2008) · Zbl 1134.65055
[28] P. Kim; P. J. Olver, Geometric integration via multi-space, Regul. Chaotic Dyn., 9, 213-226 (2004) · Zbl 1068.65092
[29] I. A. Kogan; P. J. Olver, Invariant Euler-Lagrange equations and the invariant variational bicomplex, Acta Appl. Math., 76, 137-193 (2003) · Zbl 1034.53015
[30] 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. · Zbl 1358.35072
[31] G. Marí Beffa; E. L. Mansfield, Discrete moving frames on lattice varieties and lattice-based multispaces, Found. Comput. Math., 18, 181-247 (2018) · Zbl 1390.14101
[32] R. I. McLachlan; G. R. W. Quispel; N. Robidoux, Geometric integration using discrete gradients, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357, 1021-1045 (1999) · Zbl 0933.65143
[33] T. E. Oliphant, A Guide to NumPy, Trelgol Publishing, USA, 2006.
[34] P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1993. · Zbl 0785.58003
[35] P. J. Olver, Joint invariant signatures, Found. Comput. Math., 1, 3-67 (2001) · Zbl 1001.53004
[36] P. J. Olver, Invariants of finite and discrete group actions via moving frames, preprint.
[37] P. J. Olver; J. Pohjanpelto, Moving frames for Lie pseudo-groups, Canad. J. Math., 60, 1336-1386 (2008) · Zbl 1160.53006
[38] V. Ovsienko; S. Tabachnikov, What is \(\ldots\) the Schwarzian derivative?, Notices Amer. Math. Soc., 56, 34-36 (2009) · Zbl 1176.53002
[39] G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 7pp. · Zbl 1132.65065
[40] 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. · Zbl 1396.65144
[41] R. Rebelo; F. Valiquette, Symmetry preserving numerical schemes for partial differential equations and their numerical tests, J. Difference Equ. Appl., 19, 738-757 (2103) · Zbl 1267.65111
[42] J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Applied Mathematics and Mathematical Computation, 7, Chapman & Hall, London, 1994. · Zbl 0816.65042
[43] A. T. S. Wan; A. Bihlo; J.-C. Nave, The multiplier method to construct conservative finite difference schemes for ordinary and partial differential equations, SIAM J. Numer. Anal., 54, 86-119 (2016) · Zbl 1337.65098
[44] A. T. S. Wan; A. Bihlo; J.-C. Nave, Conservative methods for dynamical systems, SIAM J. Numer. Anal., 55, 2255-2285 (2017) · Zbl 1375.65104
[45] G. Zhong; J. E. Marsden, Lie-Poisson, Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A, 133, 134-139 (1988) · Zbl 1369.70038
[46] B. Zhou and C.-J. Zhu, An application of the Schwarzian derivative, preprint, arXiv: hep-th/9907193.
[47] B. Zhou and C.-J. Zhu, The complete brane solution in \(D\)-dimensional coupled gravity system, Comm. Theor. Phys., 32 (1999).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.