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A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence

  • * Corresponding author

    * Corresponding author

The first author is supported in part by NSFC grant No.11871106, and Guangdong Natural Science Foundation through grant 2017B030311001, the third author is supported in part by NSFC grants No.11871092, U1930402.

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  • In this paper, we present and study $ C^1 $ Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree $ k $ ($ \ge 3 $) for one-dimen\-sional elliptic equations. We prove that, the solution and its derivative approximations converge with rate $ 2k-2 $ at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree $ k+1 $ in each element, the first-order derivative approximation is superconvergent at all interior $ k-2 $ Lobatto points, and the second-order derivative approximation is superconvergent at $ k-1 $ Gauss points, with an order of $ k+2 $, $ k+1 $, and $ k $, respectively. As a by-product, we prove that both the Petrov-Galerkin solution and the Gauss collocation solution are superconvergent towards a particular Jacobi projection of the exact solution in $ H^2 $, $ H^1 $, and $ L^2 $ norms. All theoretical findings are confirmed by numerical experiments.

    Mathematics Subject Classification: 65N30, 65N35, 65N12, 65N15.

    Citation:

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  • Table 1.  Errors, corresponding convergence rates for $ C^1 $ Petrov-Galerkin method, $ \alpha = \beta = \gamma = 1 $.

    $ e_{un} $ $ e_{u'n} $ $ e_u $ $ e_{u'} $ $ e_{u''} $
    $ k $ $ N $ error order error order error order error order error order
    2 8.03e-04 - 6.11e-03 - - - 7.59e-03 - 1.31e-01 -
    4 7.02e-05 3.49 4.92e-04 3.61 - - 5.61e-04 3.73 1.66e-02 2.98
    3 8 4.59e-06 3.95 3.02e-05 4.04 - - 3.73e-05 3.92 2.18e-03 2.93
    16 2.91e-07 4.04 1.90e-06 4.05 - - 2.66e-06 3.87 2.67e-04 3.03
    32 1.80e-08 4.00 1.18e-07 3.99 - - 1.77e-07 3.90 3.38e-05 2.98
    2 2.88e-05 - 2.23e-05 - 7.54e-05 - 8.72e-04 - 1.30e-02 -
    4 4.25e-07 6.10 2.47e-07 6.51 1.36e-06 5.81 2.44e-05 5.17 8.88e-04 3.88
    4 8 6.53e-09 6.21 5.38e-09 5.69 2.14e-08 6.17 7.91e-07 5.10 5.47e-05 4.02
    16 1.04e-10 5.96 1.04e-10 5.67 3.45e-10 5.94 2.64e-08 4.89 3.73e-06 3.87
    32 1.62e-12 6.00 1.84e-12 5.83 5.50e-12 5.97 8.62e-10 4.94 2.30e-07 4.02
     | Show Table
    DownLoad: CSV

    Table 2.  Errors, corresponding convergence rates for $ C^1 $ Gauss collocation method, $ \alpha = \beta = \gamma = 1 $.

    $ e_{un} $ $ e_{u'n} $ $ e_u $ $ e_{u'} $ $ e_{u^{''}} $
    $ k $ $ N $ error order error order error order error order error order
    2 5.25e-03 - 1.36e-02 - - - 1.44e-02 - 8.32e-02 -
    4 2.88e-04 4.13 7.26e-04 4.18 - - 8.35e-04 4.06 1.16e-02 2.84
    3 8 1.82e-05 4.07 4.66e-05 4.05 - - 5.89e-05 3.91 1.61e-03 2.85
    16 1.16e-06 3.94 2.91e-06 3.96 - - 4.01e-06 3.84 1.91e-04 3.08
    32 7.18e-08 4.01 1.81e-07 4.01 - - 2.65e-07 3.92 2.45e-05 2.96
    2 1.32e-05 - 1.04e-04 - 1.98e-04 - 9.54e-04 - 7.12e-03 -
    4 2.92e-07 5.48 1.79e-06 5.85 3.14e-06 5.96 3.32e-05 4.83 5.77e-04 3.63
    4 8 4.62e-09 5.97 2.80e-08 5.99 5.09e-08 5.94 1.05e-06 4.98 3.89e-05 3.89
    16 7.56e-11 6.06 4.40e-10 6.12 8.61e-10 6.01 3.40e-08 5.04 2.46e-06 3.98
    32 1.18e-12 6.08 6.87e-12 6.08 1.28e-11 6.15 1.05e-09 5.08 1.52e-07 4.02
     | Show Table
    DownLoad: CSV

    Table 3.  $ \|u_h-u_I\|_2 $ and the corresponding convergence rates, constant coefficients.

    $ \|u_h-u_I\|_2 $
    $ C^1 $ Petrov-Galerkin $ C^1 $ Gauss collocation
    $ \alpha = \beta = \gamma = 1 $ $ \alpha = \gamma = 1, \beta = 0 $ $ \alpha = \beta = \gamma = 1 $ $ \alpha = \gamma = 1, \beta = 0 $
    $ k $ $ N $ error order error order error order error order
    2 3.93e-02 - 5.57e-03 - 1.12e-01 - 8.32e-02 -
    4 5.15e-03 2.91 3.59e-04 3.94 1.36e-02 3.00 1.07e-02 2.93
    3 8 6.47e-04 3.00 2.23e-05 3.99 1.72e-03 3.04 1.34e-03 3.03
    16 8.12e-05 3.04 1.40e-06 4.02 2.17e-04 2.96 1.70e-04 2.96
    32 1.01e-05 2.99 8.75e-08 4.01 2.70e-05 3.01 2.12e-05 3.03
    2 3.47e-03 - 2.23e-04 - 9.86e-03 - 8.27e-03 -
    4 2.24e-04 3.97 7.13e-06 4.99 6.63e-04 3.88 5.20e-04 4.02
    4 8 1.40e-05 4.12 2.30e-07 5.03 4.14e-05 4.00 3.30e-05 3.94
    16 8.80e-07 3.98 7.07e-09 5.01 2.59e-06 4.08 2.07e-06 4.05
    32 5.50e-08 4.00 2.22e-10 5.02 1.62e-07 4.05 1.30e-07 4.04
     | Show Table
    DownLoad: CSV

    Table 4.  Errors and corresponding convergence rates for $ C^1 $ Petrov-Galerkin method, variable coefficients, $ k = 3 $.

    $ e_{un} $ $ e_{u'n} $ $ e_{u'} $ $ e_{u''} $
    $ k $ $ N $ error order error order error order error order
    Case 1
    4 4.24e-04 - 4.42e-04 - 1.45e-02 - 4.98e-01 -
    8 2.75e-05 3.94 2.94e-05 3.91 1.38e-03 3.39 8.75e-02 2.51
    3 16 1.75e-06 3.97 1.87e-06 3.98 1.03e-04 3.74 1.25e-02 2.81
    32 1.10e-07 4.00 1.17e-07 3.99 6.85e-06 3.91 1.63e-03 2.94
    64 6.86e-09 4.00 7.34e-09 4.00 4.36e-07 3.97 2.05e-04 2.99
    Case 2
    4 3.39e-04 - 3.42e-04 - 1.37e-02 - 4.88e-01 -
    8 2.25e-05 3.92 2.69e-05 3.67 1.31e-03 3.39 8.56e-02 2.51
    3 16 1.44e-06 3.96 1.88e-06 3.84 9.74e-05 3.74 1.22e-02 2.81
    32 9.03e-08 4.00 1.23e-07 3.93 6.46e-06 3.91 1.58e-03 2.95
    64 5.64e-09 4.00 7.87e-09 3.97 4.11e-07 3.98 2.00e-04 2.99
    Case 3
    4 3.36e-04 - 3.53e-04 - 1.37e-02 - 4.88e-01 -
    8 2.26e-05 3.89 2.82e-05 3.65 1.30e-03 3.39 8.56e-02 2.51
    3 16 1.44e-06 3.97 1.97e-06 3.84 9.72e-05 3.75 1.22e-02 2.81
    32 9.05e-08 4.00 1.29e-07 3.93 6.45e-06 3.91 1.59e-03 2.95
    64 5.66e-09 4.00 8.26e-09 3.97 4.09e-07 3.98 2.00e-04 2.99
     | Show Table
    DownLoad: CSV

    Table 5.  Errors and corresponding convergence rates for $ C^1 $ Petrov-Galerkin method, variable coefficients, $ k = 4 $.

    $ e_{un} $ $ e_{u'n} $ $ e_u $ $ e_{u'} $ $ e_{u''} $
    $ k $ $ N $ error order error order error order error order error order
    Case 1
    4 2.56e-06 - 1.71e-06 - 4.10e-05 - 1.19e-03 - 6.21e-02 -
    8 4.06e-08 5.98 3.46e-08 5.62 8.78e-07 5.55 4.83e-05 4.63 4.91e-03 3.66
    4 16 6.25e-10 6.02 6.31e-10 5.78 1.35e-08 6.03 1.38e-06 5.13 2.90e-04 4.08
    32 1.05e-11 5.89 1.03e-11 5.94 2.25e-10 5.90 4.78e-08 4.85 1.94e-05 3.90
    64 1.63e-13 6.01 1.65e-13 5.96 4.02e-12 5.81 1.52e-09 4.98 1.24e-06 3.97
    Case 2
    4 1.25e-06 - 8.07e-07 - 3.91e-05 - 1.16e-03 - 6.07e-02 -
    8 2.01e-08 5.96 2.10e-08 5.27 8.31e-07 5.56 4.62e-05 4.65 4.79e-03 3.66
    4 16 3.10e-10 6.02 3.99e-10 5.71 1.26e-08 6.05 1.33e-06 5.12 2.84e-04 4.08
    32 5.12e-12 5.92 6.70e-12 5.90 2.13e-10 5.89 4.59e-08 4.86 1.89e-05 3.90
    64 8.02e-14 6.00 1.05e-13 6.00 3.89e-12 5.77 1.46e-09 4.98 1.21e-06 3.97
    Case 3
    4 9.26e-07 - 5.75e-07 - 3.93e-05 - 1.16e-03 - 6.07e-02 -
    8 1.48e-08 5.96 1.54e-08 5.22 8.33e-07 5.56 4.62e-05 4.65 4.79e-03 3.66
    4 16 2.29e-10 6.02 2.95e-10 5.71 1.26e-08 6.05 1.33e-06 5.11 2.84e-04 4.08
    32 3.81e-12 5.91 4.94e-12 5.90 2.13e-10 5.89 4.58e-08 4.86 1.89e-05 3.90
    64 5.74e-14 6.05 9.17e-14 5.75 3.91e-12 5.77 1.46e-09 4.98 1.21e-06 3.97
     | Show Table
    DownLoad: CSV

    Table 6.  Errors and corresponding convergence rates for $ C^1 $ Gauss collocation method, variable coefficients, $ k = 3 $.

    $ e_{un} $ $ e_{u'n} $ $ e_{u'} $ $ e_{u''} $
    $ k $ $ N $ error order error order error order error order
    Case 1
    4 2.12e-03 - 3.30e-03 - 1.29e-02 - 7.15e-02 -
    8 1.43e-04 3.89 5.10e-04 2.69 1.33e-03 3.28 1.35e-02 2.41
    3 16 8.87e-06 4.01 4.71e-05 3.44 1.01e-04 3.72 2.03e-03 2.73
    32 5.49e-07 4.02 3.42e-06 3.78 6.67e-06 3.92 2.76e-04 2.88
    64 3.42e-08 4.00 2.24e-07 3.93 4.22e-07 3.98 3.59e-05 2.94
    Case 2
    4 2.30e-03 - 2.97e-03 - 1.42e-02 - 9.18e-02 -
    8 1.56e-04 3.89 5.01e-04 2.57 1.45e-03 3.30 1.71e-02 2.43
    3 16 9.62e-06 4.02 4.72e-05 3.41 1.09e-04 3.72 2.55e-03 2.74
    32 5.95e-07 4.01 3.45e-06 3.77 7.25e-06 3.92 3.46e-04 2.88
    64 3.71e-08 4.00 2.26e-07 3.93 4.59e-07 3.98 4.49e-05 2.94
    Case 3
    4 2.37e-03 - 2.84e-03 - 1.45e-02 - 9.13e-02 -
    8 1.59e-04 3.89 4.85e-04 2.55 1.47e-03 3.30 1.70e-02 2.43
    3 16 9.82e-06 4.02 4.60e-05 3.40 1.11e-04 3.73 2.54e-03 2.74
    32 6.08e-07 4.01 3.37e-06 3.77 7.33e-06 3.92 3.45e-04 2.88
    64 3.79e-08 4.00 2.21e-07 3.93 4.64e-07 3.98 4.49e-05 2.94
     | Show Table
    DownLoad: CSV

    Table 7.  Errors and corresponding convergence rates for $ C^1 $ Gauss collocation method, variable coefficients, $ k = 4 $.

    $ e_{un} $ $ e_{u'n} $ $ e_u $ $ e_{u'} $ $ e_{u''} $
    $ k $ $ N $ error order error order error order error order error order
    Case 1
    4 1.45e-05 - 1.16e-04 - 8.66e-05 - 1.00e-03 - 8.32e-03 -
    8 4.69e-07 4.95 3.01e-06 5.27 1.53e-06 5.82 3.87e-05 4.69 7.68e-04 3.44
    4 16 1.25e-08 5.23 4.84e-08 5.96 1.64e-08 6.55 1.14e-06 5.09 5.70e-05 3.75
    32 2.23e-10 5.81 7.53e-10 6.01 4.01e-10 5.35 3.76e-08 4.92 3.73e-06 3.94
    64 3.61e-12 5.95 1.18e-11 6.00 7.73e-12 5.70 1.18e-09 4.99 2.34e-07 4.00
    Case 2
    4 1.60e-05 - 1.15e-04 - 9.17e-05 - 1.09e-03 - 1.06e-02 -
    8 4.82e-07 5.05 3.09e-06 5.22 1.63e-06 5.81 4.18e-05 4.70 9.85e-04 3.42
    4 16 1.31e-08 5.20 5.06e-08 5.93 1.78e-08 6.52 1.25e-06 5.06 7.21e-05 3.77
    32 2.35e-10 5.80 7.92e-10 6.00 4.08e-10 5.45 4.10e-08 4.93 4.68e-06 3.94
    64 3.80e-12 5.95 1.24e-11 6.00 7.92e-12 5.69 1.28e-09 5.00 2.93e-07 4.00
    Case 3
    4 1.61e-05 - 1.15e-04 - 9.18e-05 - 1.09e-03 - 1.05e-02 -
    8 4.84e-07 5.06 3.08e-06 5.22 1.63e-06 5.82 4.18e-05 4.70 9.86e-04 3.42
    4 16 1.32e-08 5.20 5.07e-08 5.92 1.78e-08 6.52 1.25e-06 5.06 7.21e-05 3.77
    32 2.37e-10 5.80 7.96e-10 5.99 4.06e-10 5.45 4.10e-08 4.93 4.68e-06 3.95
    64 3.84e-12 5.95 1.25e-11 6.00 7.89e-12 5.69 1.28e-09 5.00 2.93e-07 4.00
     | Show Table
    DownLoad: CSV

    Table 8.  $ \|u_h-u_I\|_2 $ and corresponding convergence rates, variable coefficients, $ k = 3 $.

    $ \|u_h-u_I\|_2 $
    $ k $ $ N $ error order error order error order
    Case 1 Case 2 Case 3
    4 2.35e-02 - 1.89e-02 - 1.89e-02 -
    8 3.46e-03 2.77 2.82e-03 2.75 2.82e-03 2.75
    $ C^1 $ Petrov-Galerkin 3 16 4.49e-04 2.94 3.67e-04 2.94 3.67e-04 2.94
    32 5.66e-05 2.99 4.64e-05 2.99 4.64e-05 2.99
    64 7.10e-06 3.00 5.81e-06 3.00 5.81e-06 3.00
    4 1.86e-01 - 1.93e-01 - 1.93e-01 -
    8 2.65e-02 2.81 2.75e-02 2.81 2.75e-02 2.81
    $ C^1 $ Gauss collocation 3 16 3.37e-03 2.97 3.51e-03 2.97 3.51e-03 2.97
    32 4.22e-04 3.00 4.39e-04 3.00 4.39e-04 3.00
    64 5.27e-05 3.00 5.49e-05 3.00 5.49e-05 3.00
     | Show Table
    DownLoad: CSV

    Table 9.  $ \|u_h-u_I\|_2 $ and corresponding convergence rates, variable coefficients, $ k = 4 $.

    $ \|u_h-u_I\|_2 $
    $ k $ $ N $ error order error order error order
    Case 1 Case 2 Case 3
    4 5.03e-03 - 4.48e-03 - 4.48e-03 -
    8 3.53e-04 3.83 3.17e-04 3.82 3.17e-04 3.82
    $ C^1 $ Petrov-Galerkin 4 16 2.24e-05 3.98 2.01e-05 3.98 2.01e-05 3.98
    32 1.40e-06 4.00 1.26e-06 4.00 1.26e-06 4.00
    64 8.75e-08 4.00 7.86e-08 4.00 7.86e-08 4.00
    4 2.09e-02 - 2.18e-02 - 2.18e-02 -
    8 1.32e-03 3.99 1.38e-03 3.99 1.38e-03 3.99
    $ C^1 $ Gauss collocation 4 16 7.74e-05 4.09 8.12e-05 4.08 8.12e-05 4.08
    32 4.86e-06 3.99 5.09e-06 4.00 5.09e-06 4.00
    64 3.06e-07 3.99 3.20e-07 3.99 3.20e-07 3.99
     | Show Table
    DownLoad: CSV
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