BPX preconditioner for nonstandard finite element methods for diffusion problems

TL;DR

This paper introduces an optimal BPX preconditioner tailored for a broad class of finite element methods solving diffusion problems, ensuring mesh-independent efficiency and verified through numerical tests.

Contribution

It develops a unified, optimal preconditioner for various nonstandard finite element methods for diffusion, extending the BPX framework to new contexts.

Findings

Preconditioner achieves mesh-independent condition number.

Theoretical proof of optimality for various finite element methods.

Numerical experiments confirm theoretical predictions.

Abstract

This paper proposes and analyzes an optimal preconditioner for a general linear symmetric positive definite (SPD) system by following the basic idea of the well-known BPX framework. The SPD system arises from a large number of nonstandard finite element methods for diffusion problems, including the well-known hybridized Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) mixed element methods, the hybridized discontinuous Galerkin (HDG) method, the Weak Galerkin (WG) method, and the nonconforming Crouzeix-Raviart (CR) element method. We prove that the presented preconditioner is optimal, in the sense that the condition number of the preconditioned system is independent of the mesh size. Numerical experiments are provided to confirm the theoretical results.