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December  2020, 28(4): 1487-1501. doi: 10.3934/era.2020078

A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction

Zexuan Liu 1, , Zhiyuan Sun 2, and  Jerry Zhijian Yang 1,,

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
2. 

Institute of Applied Physics and Computational Mathematics, Beijing 100094, China

* Corresponding author: Jerry Zhijian Yang

Received  March 2020 Revised  June 2020 Published  July 2020

We numerically investigate the superconvergence property of the discontinuous Galerkin method by patch reconstruction. The convergence rate $ 2m+1 $ can be observed at the grid points and barycenters in one dimensional case with uniform partitions. The convergence rate $ m + 2 $ is achieved at the center of the element faces in two and three dimensions. The meshes are uniformly partitioned into triangles/tetrahedrons or squares/hexahedrons. We also demonstrate the details of the implementation of the proposed method. The numerical results for all three dimensional cases are presented to illustrate the propositions.

Keywords: Superconvergence, discontinuous Galerkin method, patch reconstruction, elliptic problem, numerical study.
Mathematics Subject Classification: Primary: 49N45; Secondary: 65N21.
Citation: Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078
References:
[1]

M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 2000. doi: 10.1002/9781118032824.  Google Scholar

[2]

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742-760.  doi: 10.1137/0719052.  Google Scholar

[3]

M. Bakker, One-dimensional Galerkin methods and superconvergence at interior nodal points, SIAM J. Numer. Anal., 21 (1984), 101-110.  doi: 10.1137/0721006.  Google Scholar

[4]

W. Cao, C.-W. Shu, Y. Yang and Z. Zhang, Superconvergence of discontinuous Galerkin methods for two-dimensional hyperbolic equations, SIAM J. Numer. Anal., 53 (2015), 1651-1671.  doi: 10.1137/140996203.  Google Scholar

[5]

P. Castillo, A superconvergence result for discontinuous Galerkin methods applied to elliptic problems, Comput. Methods Appl. Mech. Engrg., 192 (2003), 4675-4685.  doi: 10.1016/S0045-7825(03)00445-6.  Google Scholar

[6]

B. Cockburn, J. Guzmán and H. Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comp., 78 (2009), 1-24.  doi: 10.1090/S0025-5718-08-02146-7.  Google Scholar

[7]

B. Cockburn, G. Kanschat, I. Perugia and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 39 (2001), 264-285.  doi: 10.1137/S0036142900371544.  Google Scholar

[8]

B. Cockburn, W. Qiu and K. Shi, Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comp., 81 (2012), 1327-1353.  doi: 10.1090/S0025-5718-2011-02550-0.  Google Scholar

[9]

B. Cockburn, W. Qiu and K. Shi, Superconvergent HDG methods on isoparametric elements for second-order elliptic problems, SIAM J. Numer. Anal., 50 (2012), 1417-1432.  doi: 10.1137/110840790.  Google Scholar

[10]

J. Douglas Jr. and T. Dupont, Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary problems, in Topics in Numerical Analysis, Academic Press, London, 1973, 89–92.  Google Scholar

[11]

R. Li, P. Ming, Z. Sun, F. Yang and Z. Yang, A discontinuous Galerkin method by patch reconstruction for biharmonic problem, J. Comput. Math., 37 (2019), 563-580.  doi: 10.4208/jcm.1807-m2017-0276.  Google Scholar

[12]

R. Li, P. Ming, Z. Sun and Z. Yang, An arbitrary-order discontinuous Galerkin method with one unknown per element, J. Sci. Comput., 80 (2019), 268-288.  doi: 10.1007/s10915-019-00937-y.  Google Scholar

[13]

R. Li, Z. Sun, F. Yang and Z. Yang, A finite element method by patch reconstruction for the Stokes problem using mixed formulations, J. Comput. Appl. Math., 353 (2019), 1-20.  doi: 10.1016/j.cam.2018.12.017.  Google Scholar

[14]

R. Li, Z. Sun and Z. Yang, A discontinuous Galerkin method for Stokes equation by divergencefree patch reconstruction, Numer. Methods Partial Differential Equations, 36 (2020), 756-771.  doi: 10.1002/num.22449.  Google Scholar

[15]

R. Li, Z. Sun and F. Yang, Solving eigenvalue problems in a discontinuous approximation space by patch reconstruction, SIAM J. Sci. Comput., 41 (2019), A3381–A3400. doi: 10.1137/19M123693X.  Google Scholar

[16]

R. Li and F. Yang, A least squares method for linear elasticity using a patch reconstructed space, Comput. Methods Appl. Mech. Engrg., 363 (2020), 19pp. doi: 10.1016/j.cma.2020.112902.  Google Scholar

[17]

R. Li and F. Yang, A sequential least squares method for Poisson equation using a patch reconstructed space, SIAM J. Numer. Anal., 58 (2020), 353-374.  doi: 10.1137/19M1239593.  Google Scholar

[18]

R. Lin and Z. Zhang, Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems, Appl. Math., 54 (2009), 251-266.  doi: 10.1007/s10492-009-0016-6.  Google Scholar

[19]

Z. Sun, J. Liu and P. Wang, A discontinuous Galerkin method by patch reconstruction for convection-diffusion problems, Adv. Appl. Math. Mech., 12 (2020), 729-747.  doi: 10.4208/aamm.OA-2019-0193.  Google Scholar

[20]

B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, J. Comput. Phys., 135 (1997), 227-248.  doi: 10.1016/0021-9991(79)90145-1.  Google Scholar

[21]

L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics, 1605, Springer-Verlag, Berlin, 1995. doi: 10.1007/BFb0096835.  Google Scholar

[22]

J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115.  doi: 10.1016/j.cam.2012.10.003.  Google Scholar

[23]

R. Wang, R. Zhang, X. Zhang and Z. Zhang, Supercloseness analysis and polynomial preserving recovery for a class of weak Galerkin methods, Numer. Methods Partial Differential Equations, 34 (2018), 317-335.  doi: 10.1002/num.22201.  Google Scholar

[24]

Z. Xie, Z. Zhang and Z. Zhang, A numerical study of uniform superconvergence of LDG method for solving singularly perturbed problems, J. Comput. Math., 27 (2009), 280-298.   Google Scholar

[25]

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.  Google Scholar

[26]

O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg., 33 (1992), 1331-1364.  doi: 10.1002/nme.1620330702.  Google Scholar

show all references

References:
[1]

M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 2000. doi: 10.1002/9781118032824.  Google Scholar

[2]

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742-760.  doi: 10.1137/0719052.  Google Scholar

[3]

M. Bakker, One-dimensional Galerkin methods and superconvergence at interior nodal points, SIAM J. Numer. Anal., 21 (1984), 101-110.  doi: 10.1137/0721006.  Google Scholar

[4]

W. Cao, C.-W. Shu, Y. Yang and Z. Zhang, Superconvergence of discontinuous Galerkin methods for two-dimensional hyperbolic equations, SIAM J. Numer. Anal., 53 (2015), 1651-1671.  doi: 10.1137/140996203.  Google Scholar

[5]

P. Castillo, A superconvergence result for discontinuous Galerkin methods applied to elliptic problems, Comput. Methods Appl. Mech. Engrg., 192 (2003), 4675-4685.  doi: 10.1016/S0045-7825(03)00445-6.  Google Scholar

[6]

B. Cockburn, J. Guzmán and H. Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comp., 78 (2009), 1-24.  doi: 10.1090/S0025-5718-08-02146-7.  Google Scholar

[7]

B. Cockburn, G. Kanschat, I. Perugia and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 39 (2001), 264-285.  doi: 10.1137/S0036142900371544.  Google Scholar

[8]

B. Cockburn, W. Qiu and K. Shi, Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comp., 81 (2012), 1327-1353.  doi: 10.1090/S0025-5718-2011-02550-0.  Google Scholar

[9]

B. Cockburn, W. Qiu and K. Shi, Superconvergent HDG methods on isoparametric elements for second-order elliptic problems, SIAM J. Numer. Anal., 50 (2012), 1417-1432.  doi: 10.1137/110840790.  Google Scholar

[10]

J. Douglas Jr. and T. Dupont, Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary problems, in Topics in Numerical Analysis, Academic Press, London, 1973, 89–92.  Google Scholar

[11]

R. Li, P. Ming, Z. Sun, F. Yang and Z. Yang, A discontinuous Galerkin method by patch reconstruction for biharmonic problem, J. Comput. Math., 37 (2019), 563-580.  doi: 10.4208/jcm.1807-m2017-0276.  Google Scholar

[12]

R. Li, P. Ming, Z. Sun and Z. Yang, An arbitrary-order discontinuous Galerkin method with one unknown per element, J. Sci. Comput., 80 (2019), 268-288.  doi: 10.1007/s10915-019-00937-y.  Google Scholar

[13]

R. Li, Z. Sun, F. Yang and Z. Yang, A finite element method by patch reconstruction for the Stokes problem using mixed formulations, J. Comput. Appl. Math., 353 (2019), 1-20.  doi: 10.1016/j.cam.2018.12.017.  Google Scholar

[14]

R. Li, Z. Sun and Z. Yang, A discontinuous Galerkin method for Stokes equation by divergencefree patch reconstruction, Numer. Methods Partial Differential Equations, 36 (2020), 756-771.  doi: 10.1002/num.22449.  Google Scholar

[15]

R. Li, Z. Sun and F. Yang, Solving eigenvalue problems in a discontinuous approximation space by patch reconstruction, SIAM J. Sci. Comput., 41 (2019), A3381–A3400. doi: 10.1137/19M123693X.  Google Scholar

[16]

R. Li and F. Yang, A least squares method for linear elasticity using a patch reconstructed space, Comput. Methods Appl. Mech. Engrg., 363 (2020), 19pp. doi: 10.1016/j.cma.2020.112902.  Google Scholar

[17]

R. Li and F. Yang, A sequential least squares method for Poisson equation using a patch reconstructed space, SIAM J. Numer. Anal., 58 (2020), 353-374.  doi: 10.1137/19M1239593.  Google Scholar

[18]

R. Lin and Z. Zhang, Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems, Appl. Math., 54 (2009), 251-266.  doi: 10.1007/s10492-009-0016-6.  Google Scholar

[19]

Z. Sun, J. Liu and P. Wang, A discontinuous Galerkin method by patch reconstruction for convection-diffusion problems, Adv. Appl. Math. Mech., 12 (2020), 729-747.  doi: 10.4208/aamm.OA-2019-0193.  Google Scholar

[20]

B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, J. Comput. Phys., 135 (1997), 227-248.  doi: 10.1016/0021-9991(79)90145-1.  Google Scholar

[21]

L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics, 1605, Springer-Verlag, Berlin, 1995. doi: 10.1007/BFb0096835.  Google Scholar

[22]

J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115.  doi: 10.1016/j.cam.2012.10.003.  Google Scholar

[23]

R. Wang, R. Zhang, X. Zhang and Z. Zhang, Supercloseness analysis and polynomial preserving recovery for a class of weak Galerkin methods, Numer. Methods Partial Differential Equations, 34 (2018), 317-335.  doi: 10.1002/num.22201.  Google Scholar

[24]

Z. Xie, Z. Zhang and Z. Zhang, A numerical study of uniform superconvergence of LDG method for solving singularly perturbed problems, J. Comput. Math., 27 (2009), 280-298.   Google Scholar

[25]

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.  Google Scholar

[26]

O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg., 33 (1992), 1331-1364.  doi: 10.1002/nme.1620330702.  Google Scholar

Figure 2.1.  The uniform triangle mesh and the appropriate patch(left)/ the inappropriate patch(right)
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Figure 3.1.  The way to refine the mesh
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Figure 3.2.  Uniform square mesh
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Figure 3.3.  The uniform tetrahedron mesh (left)/ and the hexahedron mesh(right)
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Figure 4.1.  The sparsity patterns of the linear systems: The linear reconstruction with $ 7 $ patch size(left)/The linear reconstruction with $ 16 $ patch size(middle)/The quadratic reconstruction with $ 16 $ patch size(right)
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Figure 4.2.  The convergence order of $ \|u-\mathcal{{R}}u_h\|_{L^2(\Omega)} $(left)/$ \|\,{u-\mathcal{{R}}u_h}\,\|_{{h}} $(middle)/$ |u-\mathcal{{R}}u_h|_{h} $(right) with different order $ m $ in 1D
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Figure 4.3.  The convergence order of $ \|u-\mathcal{{R}}u_h\|_{L^2(\Omega)} $(left)/$ \|\,{u-\mathcal{{R}}u_h}\,\|_{{h}} $(right) with different order $ m $ in 2D triangle mesh
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Figure 4.4.  The convergence order of $ \|u-\mathcal{{R}}u_h\|_{L^2(\Omega)} $(left)/$ \|\,{u-\mathcal{{R}}u_h}\,\|_{{h}} $(middle))/$ |u-\mathcal{{R}}u_h|_{h} $(right) with different order $ m $ in 2d square mesh
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Figure 4.5.  The convergence order of hexahedron mesh(left) / tetrahedron mesh(right) of linear reconstruction with $ L^2 $ norm and $ |\cdot|_{h} $ quantity in 3D
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Table 4.1.  The convergence order of the different norms and quantity in 1D
$ m $ $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{L^2(\Omega)}} $ error order $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{h}} $ error order $ |u-\mathcal{{R}}u_h|_{h} $ error order
1 1.9603 2.9605 3.0536
2 3.2727 5.0225 5.0127
3 4.2114 6.8449 6.8847
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$ m $ $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{L^2(\Omega)}} $ error order $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{h}} $ error order $ |u-\mathcal{{R}}u_h|_{h} $ error order
1 1.9603 2.9605 3.0536
2 3.2727 5.0225 5.0127
3 4.2114 6.8449 6.8847
Table 4.2.  The convergence rate of different norms in 2D triangle mesh
$ m $ $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{L^2(\Omega)}} $ error order $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{h}} $ error order
1 1.9841 3.1221
2 3.3599 4.2205
3 4.0463 4.9108
4 5.2886 5.8989
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$ m $ $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{L^2(\Omega)}} $ error order $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{h}} $ error order
1 1.9841 3.1221
2 3.3599 4.2205
3 4.0463 4.9108
4 5.2886 5.8989
Table 4.3.  The convergence rate of different norms and quantity in 2D square mesh
$ m $ $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{L^2(\Omega)}} $ error order $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{h}} $ error order $ |u-\mathcal{{R}}u_h|_{h} $ error order
1 2.1375 2.9830 2.9666
2 3.0613 3.9890 3.9863
3 4.2076 4.8476 4.9693
4 4.9222 6.0021 6.0122
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$ m $ $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{L^2(\Omega)}} $ error order $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{h}} $ error order $ |u-\mathcal{{R}}u_h|_{h} $ error order
1 2.1375 2.9830 2.9666
2 3.0613 3.9890 3.9863
3 4.2076 4.8476 4.9693
4 4.9222 6.0021 6.0122
Table 4.4.  The convergence order of linear reconstruction with $ L^2 $ norm and $ |\cdot|_{h} $ quantity in 3D mesh
Mesh type $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{L^2(\Omega)}} $ error order $ |u-\mathcal{{R}}u_h|_{h} $ error order
Tetrahedron 1.9468 3.0814
Hexahedron 2.1064 3.0191
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Mesh type $ \|\,{u-\mathcal{{R}}u_h}\,\|_{{L^2(\Omega)}} $ error order $ |u-\mathcal{{R}}u_h|_{h} $ error order
Tetrahedron 1.9468 3.0814
Hexahedron 2.1064 3.0191
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