Supercloseness analysis and polynomial preserving recovery for a class of weak Galerkin method

TL;DR

This paper investigates the supercloseness and superconvergence properties of weak Galerkin finite element methods for elliptic problems, introducing stabilization and polynomial preserving recovery techniques to enhance solution accuracy.

Contribution

It presents a novel supercloseness analysis for WG methods and develops a PPR-based post-processing technique for improved superconvergence.

Findings

WG solutions are superclose to Lagrange interpolation with Lobatto points

New stabilization terms enable supercloseness analysis

PPR post-processing achieves superconvergence in numerical tests

Abstract

In this paper, 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 type interpolation using Lobatto points. This supercloseness behavior is obtained through some newly designed stabilization terms. A post-processing technique using the polynomial preserving recovery (PPR) is introduced for WG approximation. Superconvergence analysis is carried out for the PPR approximation. Numerical examples are provided to verify our theoretical results.