A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction

Abstract

<p style='text-indent:20px;'>We numerically investigate the superconvergence property of the discontinuous Galerkin method by patch reconstruction. The convergence rate <inline-formula><tex-math id="M1">$ 2m+1 $</tex-math></inline-formula> can be observed at the grid points and barycenters in one dimensional case with uniform partitions. The convergence rate <inline-formula><tex-math id="M2">$ m + 2 $</tex-math></inline-formula> 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.</p>