Supercloseness analysis and polynomial preserving Recovery for a class of weak Galerkin Methods

Abstract

In this article, we analyze convergence and supercloseness properties of a class of weak Galerkin (WG) finite element methods for solving second-order elliptic problems. It is shown that the WG solution is superclose to the Lagrange interpolant using Lobatto points. This supercloseness behavior is obtained through some newly designed stabilization terms. A postprocessing technique using polynomial preserving recovery (PPR) is introduced for the WG approximation. Superconvergence analysis is performed for the PPR recovered gradient. Numerical examples are provided to illustrate our theoretical results. K E Y W O R D Ssecond-order elliptic equation, polynomial preserving recovery, supercloseness, superconvergence, weak Galerkin method Numer Methods Partial Differential Eq. 2018;34:317-335.wileyonlinelibrary.com/journal/num

Superconvergence of the gradient approximation for weak Galerkin finite element methods on nonuniform rectangular partitions

“…It was shown in [10] that the projected numerical solution of the lowest order is convergent to the exact solution at the rate of O(h 1.5 ) or better in the usual H 1 -norm. In [36], a post-processing technique using the polynomial preserving recovery (PPR) was introduced for the WG approximation arising from schemes with bi-polynomials and over-penalized stabilization terms on uniform rectangular partitions.…”

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

“…The weak Galerkin(WG) method was introduced by Wang and Ye in [22], and also observed a superconvergence properties numerically. In [23], Wang et al analyzed the superconvergence for the polynomial preserving recovered gradient of the WG method.…”

Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions

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