MultiGrid Preconditioners for Mixed Finite Element Methods of the Vector Laplacian

Long Chen,Jie Zhou,Lin Zhong,Yongke Wu

doi:10.1007/s10915-018-0697-7

Abstract

Due to the indefiniteness and poor spectral properties, the discretized linear algebraic system of the vector Laplacian by mixed finite element methods is hard to solve. A block diagonal preconditioner has been developed and shown to be an effective preconditioner by Arnold et al. (Acta Numer 15:1–155, 2006). The purpose of this paper is to propose alternative and effective block diagonal and approximate block factorization preconditioners for solving these saddle point systems. A variable V-cycle multigrid method with the standard point-wise Gauss–Seidel smoother is proved to be a good preconditioner for the discrete vector Laplacian operator. The major benefit of our approach is that the point-wise Gauss–Seidel smoother is more algebraic and can be easily implemented as a black-box smoother. This multigrid solver will be further used to build preconditioners for the saddle point systems of the vector Laplacian. Furthermore it is shown that Maxwell’s equations with the divergent free constraint can be decoupled into one vector Laplacian and one scalar Laplacian equation.

Highlights

Discretization of the vector Laplacian in spaces H0(curl ) and H0(div) by mixed finite element methods is well-studied in [1]
With the approximation and smoothing properties, we show that one V-cycle is an effective preconditioner
We propose a block diagonal preconditioner and a block triangular preconditioner: (3)

Summary

Introduction

Discretization of the vector Laplacian in spaces H0(curl ) and H0(div) by mixed finite element methods is well-studied in [1]. The discretized linear algebraic system is ill-conditioned and in the saddle point form which leads to the slow convergence of classical iterative methods as the size of the system becomes large. In [1], a block diagonal preconditioner has been developed and shown to be an effective preconditioner. The purpose of this paper is to present alternative and effective block diagonal and block triangular preconditioners for solving these saddle point systems. Due to the similarity of the problems arising from spaces H0(curl ) and H0(div), we use the mixed formulation of the vector Laplacian in H0(curl ) as an example to illustrate our approach. Choosing appropriate finite element spaces Sh ⊂ H01 (a vertex element space) and U h ⊂ H0(curl ) (an edge element space), the mixed formulation is: Find σh ∈ Sh, uh ∈ U h such that

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