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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  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.

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
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##### 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
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