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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

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.

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. CaoC.-W. ShuY. 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

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B. CockburnJ. 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

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B. CockburnG. KanschatI. 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

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B. CockburnW. 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. CockburnW. 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. LiP. MingZ. SunF. 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. LiP. MingZ. 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. LiZ. SunF. 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. LiZ. 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. SunJ. 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. WangR. ZhangX. 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. XieZ. 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. CaoC.-W. ShuY. 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. CockburnJ. 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. CockburnG. KanschatI. 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. CockburnW. 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. CockburnW. 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. LiP. MingZ. SunF. 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. LiP. MingZ. 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. LiZ. SunF. 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. LiZ. 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. SunJ. 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. WangR. ZhangX. 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. XieZ. 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)
Figure 3.1.  The way to refine the mesh
Figure 3.2.  Uniform square mesh
Figure 3.3.  The uniform tetrahedron mesh (left)/ and the hexahedron mesh(right)
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)
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
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
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
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
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
$ 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
$ 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
$ 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
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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