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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
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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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