American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium
  • ERA Home
  • This Issue
  • Next Article
    Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems
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  December 2020 Early access  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)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.1.  The way to refine the mesh
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.2.  Uniform square mesh
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.3.  The uniform tetrahedron mesh (left)/ and the hexahedron mesh(right)
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

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
[1]

Mahboub Baccouch. Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 19-54. doi: 10.3934/dcdsb.2018104
[2]

Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic & Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139
[3]

Waixiang Cao, Lueling Jia, Zhimin Zhang. A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327
[4]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
[5]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319
[6]

Qingjie Hu, Zhihao Ge, Yinnian He. Discontinuous Galerkin method for the Helmholtz transmission problem in two-level homogeneous media. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2923-2948. doi: 10.3934/dcdsb.2020046
[7]

Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817
[8]

Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial & Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631
[9]

Tiexiang Li, Tsung-Ming Huang, Wen-Wei Lin, Jenn-Nan Wang. On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method. Inverse Problems & Imaging, 2018, 12 (4) : 1033-1054. doi: 10.3934/ipi.2018043
[10]

Zhen-Zhen Tao, Bing Sun. Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021220
[11]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, 2021, 29 (3) : 2375-2389. doi: 10.3934/era.2020120
[12]

José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini, Gershon Wolansky. Numerical study of a particle method for gradient flows. Kinetic & Related Models, 2017, 10 (3) : 613-641. doi: 10.3934/krm.2017025
[13]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768
[14]

Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
[15]

Luis J. Roman, Marcus Sarkis. Stochastic Galerkin method for elliptic spdes: A white noise approach. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 941-955. doi: 10.3934/dcdsb.2006.6.941
[16]

ShinJa Jeong, Mi-Young Kim. Computational aspects of the multiscale discontinuous Galerkin method for convection-diffusion-reaction problems. Electronic Research Archive, 2021, 29 (2) : 1991-2006. doi: 10.3934/era.2020101
[17]

Kim S. Bey, Peter Z. Daffer, Hideaki Kaneko, Puntip Toghaw. Error analysis of the p-version discontinuous Galerkin method for heat transfer in built-up structures. Communications on Pure & Applied Analysis, 2007, 6 (3) : 719-740. doi: 10.3934/cpaa.2007.6.719
[18]

Xia Ji, Wei Cai. Accurate simulations of 2-D phase shift masks with a generalized discontinuous Galerkin (GDG) method. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 401-415. doi: 10.3934/dcdsb.2011.15.401
[19]

Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic & Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033
[20]

Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489

2020 Impact Factor: 1.833

Tools

Metrics

  • PDF downloads (367)
  • HTML views (582)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy