An auxiliary space multigrid preconditioner for the weak Galerkin method

TL;DR

This paper develops an auxiliary space multigrid preconditioner for the weak Galerkin finite element method, improving solver efficiency for second-order diffusion equations on simplicial meshes with mesh-independent condition numbers.

Contribution

The authors introduce a novel auxiliary space multigrid preconditioner for the weak Galerkin method, linking it with $H^1$ conforming spaces to enhance computational performance.

Findings

Preconditioned systems have condition numbers independent of mesh size.

Numerical experiments confirm the theoretical effectiveness of the preconditioner.

The method simplifies solving reduced systems involving edge/face degrees of freedom.

Abstract

In this paper, the authors constructed an auxiliary space multigrid preconditioner for the weak Galerkin finite element method for second-order diffusion equations, discretized on simplicial 2D or 3D meshes. The idea of the auxiliary space multigrid preconditioner is to use an auxiliary space as a "coarse" space in the multigrid algorithm, where the discrete problem in the auxiliary space can be easily solved by an existing solver. In this construction, the authors conveniently use the $H^1$ conforming piecewise linear finite element space as an auxiliary space. The main technical difficulty is to build the connection between the weak Galerkin discrete space and the $H^1$ conforming piecewise linear finite element space. The authors successfully constructed such an auxiliary space multigrid preconditioner for the weak Galerkin method, as well as a reduced system of the weak Galerkin method involving only the degrees of freedom on edges/faces. The preconditioned systems are proved to have condition numbers independent of the mesh size. Numerical experiments are conducted to support the theoretical results.