A two-level algorithm for the weak Galerkin discretization of diffusion problems

TL;DR

This paper presents a two-level algorithm for weak Galerkin finite element methods applied to diffusion problems, demonstrating optimal condition numbers and convergence without regularity assumptions through theoretical analysis and numerical validation.

Contribution

It introduces a novel two-level algorithm for WG discretizations of diffusion problems, with proven convergence and optimal condition number estimates.

Findings

Condition numbers are of order O(h^{-2})

Convergence proven without regularity assumptions

Numerical results confirm theoretical analysis

Abstract

This paper analyzes a two-level algorithm for the weak Galerkin (WG) finite element methods based on local Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) mixed elements for two- and three-dimensional diffusion problems with Dirichlet condition. We first show the condition numbers of the stiffness matrices arising from the WG methods are of $O(h^{-2})$. We use an extended version of the Xu-Zikatanov (XZ) identity to derive the convergence of the algorithm without any regularity assumption. Finally we provide some numerical results.