New Preconditioners with Two Variable Relaxation Parameters for the Discretized Time‐Harmonic Maxwell Equations in Mixed Form

Yuping Zeng,Chenliang Li

doi:10.1155/2012/617390

Abstract

We provide new preconditioners with two variable relaxation parameters for the saddle point linear systems arising from finite element discretization of time‐harmonic Maxwell equations in mixed form. The new preconditioners are of block‐triangular forms and Schur complement‐free. They are extensions of the results in Cheng et al., 2009, Grief and Schötzau, 2007, and Huang et al., 2009. Theoretical analysis shows that all eigenvalues of the preconditioned matrices are tightly clustered, and numerical tests confirm our analysis.

Highlights

We consider the preconditioning techniques for solving the saddle point linear systems arising from finite element discretization of the following time-harmonic Maxwell equations in mixed form 1–5 : find the vector field u and the Lagrangian multiplier p such that∇ × ∇ × u − k2u ∇p f in Ω, ∇ · u 0 in Ω, 1.1 u × n 0 on ∂Ω, p 0 on ∂Ω.Here, Ω ⊂ R2 is a connected polyhedron domain with a connected boundary ∂Ω, and n denotes the outward unit normal on ∂Ω
We provide new preconditioners with two variable relaxation parameters for the saddle point linear systems arising from finite element discretization of time-harmonic Maxwell equations in mixed form
We consider the saddle-point linear system 1.2 arising from the discretized time-harmonic Maxwell equations in mixed form 1.1 and assume that k2 is not an eigenvalue and k2 ∈ R

Summary

Introduction

We consider the preconditioning techniques for solving the saddle point linear systems arising from finite element discretization of the following time-harmonic Maxwell equations in mixed form 1–5 : find the vector field u and the Lagrangian multiplier p such that. There have been many techniques for solving Maxwell equations, such as the geometry multigrid methods 6–8 , algebraic multigrid methods 9 , domain decomposition methods 4, 10–13 , Nodal auxiliary space preconditioning methods 14 , and the solution methods to the corresponding saddle-point linear systems 2, 3, 15. Based on the work of Grief and Schotzau 3 , 2 gives block-triangular Schur complement-free preconditioners for the linear system 1.2.

New Block-Triangular Preconditioners for Any k2

Numerical Experiments

Conclusions