American Institute of Mathematical Sciences

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January  2021, 26(1): 81-105. doi: 10.3934/dcdsb.2020327

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

Waixiang Cao 1, , Lueling Jia 2, and  Zhimin Zhang 2,,

1. 

School of Mathematical Science, Beijing Normal University, Beijing l00875, China
2. 

Beijing Computational Science Research Center, Beijing 100193, China, Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

* Corresponding author

Received  February 2020 Revised  September 2020 Published  January 2021 Early access  November 2020

Fund Project: 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

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.

Keywords: Hermite interpolation, $ C^1 $ elements, superconvergence, Gauss collocation methods, Petrov-Galerkin methods, Jacobi polynomials.
Mathematics Subject Classification: 65N30, 65N35, 65N12, 65N15.
Citation: Waixiang Cao, Lueling Jia, Zhimin Zhang. A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327
References:
[1]

S. Adjerid and T. C. Massey, Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem, Comput. Methods Appl. Mech. Engrg., 195 (2006), 3331-3346.  doi: 10.1016/j.cma.2005.06.017.

[2]

S. Adjerid and T. Weinhart, Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3113-3129.  doi: 10.1016/j.cma.2009.05.016.

[3]

S. Adjerid and T. Weinhart, Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems,, Math. Comp., 80 (2011), 1335-1367.  doi: 10.1090/S0025-5718-2011-02460-9.

[4]

I. Babu$ \rm\check{s} $ka, T. Strouboulis, C. S. Upadhyay and S. K. Gangaraj, Computer-based proof of the existence of superconvergence points in the finite element method: Superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's, and the elasticity equations, Numer. Meth. PDEs, 12 (1996), 347-392.  doi: 10.1002/num.1690120303.

[5]

S. K. Bhal and P. Danumjaya, A Fourth-order orthogonal spline collocation solution to 1D-Helmholtz equation with discontinuity, J. Anal., 27 (2019), 377-390.  doi: 10.1007/s41478-018-0082-9.

[6]

B. Bialecki, Superconvergence of the orthogonal spline collocation solution of Poisson's equation,, Numerical Methods for Partial Differential Equations, 15 (1999), 285-303.  doi: 10.1002/(SICI)1098-2426(199905)15:3<285::AID-NUM2>3.0.CO;2-1.

[7]

J. H. Bramble and A. H. Schatz, High order local accuracy by averaging in the finite element method, Math. Comp., 31 (1997), 94-111.  doi: 10.1090/S0025-5718-1977-0431744-9.

[8]

Z. Q. Cai, On the finite volume element method, Numer. Math., 58 (1991), 713-735.  doi: 10.1007/BF01385651.

[9]

W. Cao, C.-W. Shu, Y. Yang and Z. Zhang, Superconvergence of Discontinuous Galerkin method for nonlinear hyperbolic equations, SIAM. J. Numer. Anal., 56 (2018), 732-765.  doi: 10.1137/17M1128605.

[10]

W. Cao and Z. Zhang, Superconvergence of Local Discontinuous Galerkin method for one-dimensional linear parabolic equations, Math. Comp., 85 (2016), 63-84.  doi: 10.1090/mcom/2975.

[11]

W. Cao, Z. Zhang and Q. Zou, Superconvergence of any order finite volume schemes for 1D general elliptic equations, J. Sci. Comput., 56 (2013), 566-590.  doi: 10.1007/s10915-013-9691-2.

[12]

W. Cao, Z. Zhang and Q. Zou, Superconvergence of Discontinuous Galerkin method for linear hyperbolic equations, SIAM. J. Numer. Anal., 52 (2014), 2555-2573.  doi: 10.1137/130946873.

[13]

W. Cao, Z. Zhang and Q. Zou, Is $2k$-conjecture valid for finite volume methods?, SIAM. J. Numer. Anal., 53 (2015), 942-962.  doi: 10.1137/130936178.

[14] C. Chen, Structure Theory of Superconvergence of Finite Elements, Hunan Science and Technology Press, Hunan, China, 2001. 
[15]

C. Chen and S. Hu, The highest order superconvergence for bi-$k$ degree rectangular elements at nodes- a proof of $2k$-conjecture,, Math. Comp., 82 (2013), 1337-1355.  doi: 10.1090/S0025-5718-2012-02653-6.

[16] C. Chen and Y. Huang, High Accuracy Theory of Finite Elements, Hunan Science and Technology Press, Hunan, China, 1995. 
[17]

Y. Cheng and C.-W. Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM J. Numer. Anal., 47 (2010), 4044-4072.  doi: 10.1137/090747701.

[18]

S.-H. Chou and X. Ye, Superconvergence of finite volume methods for the second order elliptic problem, Comput. Methods Appl. Mech. Eng., 196 (2007), 3706-3712.  doi: 10.1016/j.cma.2006.10.025.

[19] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second edition, Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. 
[20]

R. E. Ewing, R. D. Lazarov and J. Wang, Superconvergence of the velocity along the Gauss lines in mixed finite element methods, SIAM J. Numer. Anal., 28 (1991), 1015-1029.  doi: 10.1137/0728054.

[21]

M. K$ \rm\check{r} $í$ \rm\check{z} $ek and P. Neittaanm$\ddot{a}$ki, On superconvergence techniques, Acta Appl. Math., 9 (1987), 175-198.  doi: 10.1007/BF00047538.

[22]

M. K$ \rm\check{r} $í$ \rm\check{z} $ek, P. Neittaanm$\ddot{a}$ki and R. Stenberg (Eds.), Finite Element Methods: Superconvergence, Post-processing, and A Posteriori Estimates, Lecture Notes in Pure and Applied Mathematics Series Vol. 196, Marcel Dekker, Inc., New York, 1997.

[23] Q. Lin and N. Yan, Construction and Analysis of High Efficient Finite Elements, Hebei University Press, P.R. China, 1996. 
[24]

A. H. Schatz, I. H. Sloan and L. B. Wahlbin, Superconvergence in finite element methods and meshes which are symmetric with respect to a point, SIAM J. Numer. Anal., 33 (1996), 505-521.  doi: 10.1137/0733027.

[25]

J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[26]

V. Thomée, High order local approximation to derivatives in the finite element method, Math. Comp., 31 (1997), 652-660.  doi: 10.1090/S0025-5718-1977-0438664-4.

[27]

L. B. Wahlbin, Superconvergence In Galerkin Finite Element Methods, Lecture Notes in Mathematics, 1605. Spring, Berlin, 1995. doi: 10.1007/BFb0096835.

[28]

Z. Xie and Z. Zhang, Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D, Math. Comp., 79 (2010), 35-45.  doi: 10.1090/S0025-5718-09-02297-2.

[29]

J. Xu and Q. Zou, Analysis of linear and quadratic simplitical finite volume methods for elliptic equations, Numer. Math., 111 (2009), 469-492.  doi: 10.1007/s00211-008-0189-z.

[30]

Y. Yang and C.-W. Shu, Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM J. Numer. Anal., 50 (2012), 3110-3133.  doi: 10.1137/110857647.

[31]

Z. Zhang, Superconvergence points of polynomial spectral interpolation,, SIAM J. Numer. Anal., 50 (2012), 2966-2985.  doi: 10.1137/120861291.

[32]

Z. Zhang, Superconvergence of a Chebyshev spectral collocation method, J. Sci. Comput., 34 (2008), 237-246.  doi: 10.1007/s10915-007-9163-7.

[33] Q. Zhu and Q. Lin, Superconvergence Theory of the Finite Element Method, Hunan Science and Technology Press, Hunan, China, 1989. 

show all references

References:
[1]

S. Adjerid and T. C. Massey, Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem, Comput. Methods Appl. Mech. Engrg., 195 (2006), 3331-3346.  doi: 10.1016/j.cma.2005.06.017.

[2]

S. Adjerid and T. Weinhart, Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3113-3129.  doi: 10.1016/j.cma.2009.05.016.

[3]

S. Adjerid and T. Weinhart, Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems,, Math. Comp., 80 (2011), 1335-1367.  doi: 10.1090/S0025-5718-2011-02460-9.

[4]

I. Babu$ \rm\check{s} $ka, T. Strouboulis, C. S. Upadhyay and S. K. Gangaraj, Computer-based proof of the existence of superconvergence points in the finite element method: Superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's, and the elasticity equations, Numer. Meth. PDEs, 12 (1996), 347-392.  doi: 10.1002/num.1690120303.

[5]

S. K. Bhal and P. Danumjaya, A Fourth-order orthogonal spline collocation solution to 1D-Helmholtz equation with discontinuity, J. Anal., 27 (2019), 377-390.  doi: 10.1007/s41478-018-0082-9.

[6]

B. Bialecki, Superconvergence of the orthogonal spline collocation solution of Poisson's equation,, Numerical Methods for Partial Differential Equations, 15 (1999), 285-303.  doi: 10.1002/(SICI)1098-2426(199905)15:3<285::AID-NUM2>3.0.CO;2-1.

[7]

J. H. Bramble and A. H. Schatz, High order local accuracy by averaging in the finite element method, Math. Comp., 31 (1997), 94-111.  doi: 10.1090/S0025-5718-1977-0431744-9.

[8]

Z. Q. Cai, On the finite volume element method, Numer. Math., 58 (1991), 713-735.  doi: 10.1007/BF01385651.

[9]

W. Cao, C.-W. Shu, Y. Yang and Z. Zhang, Superconvergence of Discontinuous Galerkin method for nonlinear hyperbolic equations, SIAM. J. Numer. Anal., 56 (2018), 732-765.  doi: 10.1137/17M1128605.

[10]

W. Cao and Z. Zhang, Superconvergence of Local Discontinuous Galerkin method for one-dimensional linear parabolic equations, Math. Comp., 85 (2016), 63-84.  doi: 10.1090/mcom/2975.

[11]

W. Cao, Z. Zhang and Q. Zou, Superconvergence of any order finite volume schemes for 1D general elliptic equations, J. Sci. Comput., 56 (2013), 566-590.  doi: 10.1007/s10915-013-9691-2.

[12]

W. Cao, Z. Zhang and Q. Zou, Superconvergence of Discontinuous Galerkin method for linear hyperbolic equations, SIAM. J. Numer. Anal., 52 (2014), 2555-2573.  doi: 10.1137/130946873.

[13]

W. Cao, Z. Zhang and Q. Zou, Is $2k$-conjecture valid for finite volume methods?, SIAM. J. Numer. Anal., 53 (2015), 942-962.  doi: 10.1137/130936178.

[14] C. Chen, Structure Theory of Superconvergence of Finite Elements, Hunan Science and Technology Press, Hunan, China, 2001. 
[15]

C. Chen and S. Hu, The highest order superconvergence for bi-$k$ degree rectangular elements at nodes- a proof of $2k$-conjecture,, Math. Comp., 82 (2013), 1337-1355.  doi: 10.1090/S0025-5718-2012-02653-6.

[16] C. Chen and Y. Huang, High Accuracy Theory of Finite Elements, Hunan Science and Technology Press, Hunan, China, 1995. 
[17]

Y. Cheng and C.-W. Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM J. Numer. Anal., 47 (2010), 4044-4072.  doi: 10.1137/090747701.

[18]

S.-H. Chou and X. Ye, Superconvergence of finite volume methods for the second order elliptic problem, Comput. Methods Appl. Mech. Eng., 196 (2007), 3706-3712.  doi: 10.1016/j.cma.2006.10.025.

[19] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second edition, Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. 
[20]

R. E. Ewing, R. D. Lazarov and J. Wang, Superconvergence of the velocity along the Gauss lines in mixed finite element methods, SIAM J. Numer. Anal., 28 (1991), 1015-1029.  doi: 10.1137/0728054.

[21]

M. K$ \rm\check{r} $í$ \rm\check{z} $ek and P. Neittaanm$\ddot{a}$ki, On superconvergence techniques, Acta Appl. Math., 9 (1987), 175-198.  doi: 10.1007/BF00047538.

[22]

M. K$ \rm\check{r} $í$ \rm\check{z} $ek, P. Neittaanm$\ddot{a}$ki and R. Stenberg (Eds.), Finite Element Methods: Superconvergence, Post-processing, and A Posteriori Estimates, Lecture Notes in Pure and Applied Mathematics Series Vol. 196, Marcel Dekker, Inc., New York, 1997.

[23] Q. Lin and N. Yan, Construction and Analysis of High Efficient Finite Elements, Hebei University Press, P.R. China, 1996. 
[24]

A. H. Schatz, I. H. Sloan and L. B. Wahlbin, Superconvergence in finite element methods and meshes which are symmetric with respect to a point, SIAM J. Numer. Anal., 33 (1996), 505-521.  doi: 10.1137/0733027.

[25]

J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[26]

V. Thomée, High order local approximation to derivatives in the finite element method, Math. Comp., 31 (1997), 652-660.  doi: 10.1090/S0025-5718-1977-0438664-4.

[27]

L. B. Wahlbin, Superconvergence In Galerkin Finite Element Methods, Lecture Notes in Mathematics, 1605. Spring, Berlin, 1995. doi: 10.1007/BFb0096835.

[28]

Z. Xie and Z. Zhang, Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D, Math. Comp., 79 (2010), 35-45.  doi: 10.1090/S0025-5718-09-02297-2.

[29]

J. Xu and Q. Zou, Analysis of linear and quadratic simplitical finite volume methods for elliptic equations, Numer. Math., 111 (2009), 469-492.  doi: 10.1007/s00211-008-0189-z.

[30]

Y. Yang and C.-W. Shu, Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM J. Numer. Anal., 50 (2012), 3110-3133.  doi: 10.1137/110857647.

[31]

Z. Zhang, Superconvergence points of polynomial spectral interpolation,, SIAM J. Numer. Anal., 50 (2012), 2966-2985.  doi: 10.1137/120861291.

[32]

Z. Zhang, Superconvergence of a Chebyshev spectral collocation method, J. Sci. Comput., 34 (2008), 237-246.  doi: 10.1007/s10915-007-9163-7.

[33] Q. Zhu and Q. Lin, Superconvergence Theory of the Finite Element Method, Hunan Science and Technology Press, Hunan, China, 1989. 
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
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$ 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
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
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$ 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
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
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$ \|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
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
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$ 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
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
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$ 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
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
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$ 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
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
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$ 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
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
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$ \|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
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
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$ \|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
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