Supercloseness analysis and polynomial preserving recovery for a class of weak Galerkin method
For researchers in numerical PDEs, this provides a theoretical foundation for supercloseness and superconvergence in weak Galerkin methods, though the results are incremental.
The paper proves that the weak Galerkin finite element solution is superclose to Lagrange interpolation at Lobatto points for second-order elliptic problems, and introduces a polynomial preserving recovery technique that achieves superconvergence. Numerical examples confirm the theoretical results.
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.