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

Alex Bihlo; James Jackaman; Francis Valiquette

doi:10.3934/jcd.2020014

Article Contents

2020, Volume 7, Issue 2: 339-368. Doi: 10.3934/jcd.2020014

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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
^* Corresponding author: James Jackaman

Received: June 30, 2019

Published: July 2020

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

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 65L60, 34C14; Secondary: 58D19.

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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 $

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Figure 2. Absolute difference between the exact solution (110) and the naive discretisation (104) and the invariant discretisation (108) with $ {\tau}{} = 0.01 $

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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

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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

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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

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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

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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

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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

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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	×	×

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