# American Institute of Mathematical Sciences

• Previous Article
Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation
• ERA Home
• This Issue
• Next Article
Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium
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, doi: 10.3934/era.2020078
##### References:

show all references

##### References:
The uniform triangle mesh and the appropriate patch(left)/ the inappropriate patch(right)
The way to refine the mesh
Uniform square mesh
The uniform tetrahedron mesh (left)/ and the hexahedron mesh(right)
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)
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
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
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
The convergence order of hexahedron mesh(left) / tetrahedron mesh(right) of linear reconstruction with $L^2$ norm and $|\cdot|_{h}$ quantity in 3D
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
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
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
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
 [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] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185 [16] 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 [17] Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503 [18] Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473 [19] Tobias Breiten, Karl Kunisch, Laurent Pfeiffer. Numerical study of polynomial feedback laws for a bilinear control problem. Mathematical Control & Related Fields, 2018, 8 (3&4) : 557-582. doi: 10.3934/mcrf.2018023 [20] Lunji Song, Zhimin Zhang. Polynomial preserving recovery of an over-penalized symmetric interior penalty Galerkin method for elliptic problems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1405-1426. doi: 10.3934/dcdsb.2015.20.1405

2018 Impact Factor: 0.263

## Tools

Article outline

Figures and Tables