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Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics, 1605, Springer-Verlag, Berlin, 1995.
doi: 10.1007/BFb0096835. |
[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. |
[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. |
[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.
|
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics, 1605, Springer-Verlag, Berlin, 1995.
doi: 10.1007/BFb0096835. |
[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. |
[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. |
[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.
|
[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. |
[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. |









1 | 1.9603 | 2.9605 | 3.0536 |
2 | 3.2727 | 5.0225 | 5.0127 |
3 | 4.2114 | 6.8449 | 6.8847 |
1 | 1.9603 | 2.9605 | 3.0536 |
2 | 3.2727 | 5.0225 | 5.0127 |
3 | 4.2114 | 6.8449 | 6.8847 |
1 | 1.9841 | 3.1221 |
2 | 3.3599 | 4.2205 |
3 | 4.0463 | 4.9108 |
4 | 5.2886 | 5.8989 |
1 | 1.9841 | 3.1221 |
2 | 3.3599 | 4.2205 |
3 | 4.0463 | 4.9108 |
4 | 5.2886 | 5.8989 |
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 |
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 |
Mesh type | ||
Tetrahedron | 1.9468 | 3.0814 |
Hexahedron | 2.1064 | 3.0191 |
Mesh type | ||
Tetrahedron | 1.9468 | 3.0814 |
Hexahedron | 2.1064 | 3.0191 |
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