\`x^2+y_1+z_12^34\`
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On the development of symmetry-preserving finite element schemes for ordinary differential equations

  • * Corresponding author: James Jackaman

    * Corresponding author: James Jackaman 

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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  • 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.

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

    Citation:

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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 $

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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 × ×
     | Show Table
    DownLoad: CSV
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