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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 617390, 13 pages
http://dx.doi.org/10.1155/2012/617390
Research Article

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

1Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Jiangsu, Nanjing 210046, China
2School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guangxi, Guilin 541004, China

Received 4 May 2012; Revised 3 July 2012; Accepted 4 July 2012

Academic Editor: Ion Zaballa

Copyright © 2012 Yuping Zeng and Chenliang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. 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 [15]: find the vector field 𝑢 and the Lagrangian multiplier 𝑝 such that ××𝑢𝑘2𝑢+𝑝=𝑓inΩ,𝑢=0inΩ,𝑢×𝑛=0on𝜕Ω,𝑝=0on𝜕Ω.(1.1) Here, Ω2 is a simply connected polyhedron domain with a connected boundary 𝜕Ω, and 𝑛 denotes the outward unit normal on 𝜕Ω. The datum 𝑓 is a given source (not necessarily divergence-free), and the wave number 𝑘2=𝜔2𝜖𝜇, where 𝜔0 is the frequency, and 𝜖 and 𝜇 are positive permittivity and permeability parameters, respectively.

In recent years, there have been many techniques for solving Maxwell equations, such as the geometry multigrid methods [68], algebraic multigrid methods [9], domain decomposition methods [4, 1013], Nodal auxiliary space preconditioning methods [14], and the solution methods to the corresponding saddle-point linear systems [2, 3, 15]. We can also use Uzawa-type iterative methods [16, 17] and preconditioned Krylov subspace methods [1824] to solve the saddle-point linear systems. Based on the previous works in [2, 3, 15], we will further study solution methods for the saddle-point linear systems in this paper.

Using Nédélec elements of the first kind [2527] for the approximation of the vector field and standard nodal elements for the Lagrangian multiplier yields the following saddle-point linear system: 𝒜𝑥𝐴𝑘2𝑀𝐵𝑇𝑢𝑝=𝑔0𝐵0𝑏,(1.2) where 𝑢𝐑𝑛 and 𝑝𝐑𝑚 are finite arrays, and 𝑔𝐑𝑛 is a load vector associated with 𝑓. The matrix 𝐴𝐑𝑛×𝑛 is symmetric positive semidefinite with nullity 𝑚 and corresponds to the curl-curl operator; 𝐵𝐑𝑚×𝑛 is a discrete divergence operator with full-row rank, and 𝑀𝐑𝑛×𝑛 is a vector mass matrix.

For convenience, we denote the standard Euclidean inner product of vectors by , and the null space of a matrix by null (). For a given positive (semi)definite matrix 𝑊 and a vector 𝑥, we define the (semi)norm: |𝑥|𝑊=𝑊𝑥,𝑥.(1.3)

The matrices 𝐴 and 𝐵 have the following stability properties [3]. Let 𝐴𝑢,𝑢=|𝑢|2𝐴. Then there exists an 𝛼, 0<𝛼<1, such that |𝑢|2𝐴𝛼|𝑢|2𝑀,𝑢null(𝐵),(1.4) where 𝛼=𝛼/(1𝛼). Matrix 𝐵 satisfies the discrete inf-sup condition: inf0𝑞𝐑𝑚sup0𝑣null(𝐴)𝐵𝑣,𝑞|𝑣|𝑀||𝑞||𝐿𝛽>0,(1.5) where the inf-sup constant 𝛽>0 is only dependent on the domain Ω.

If the wave number 𝑘2>0, then the (1,1) block of (1.2) is indefinite. For difficulty and corresponding solution methods of this problem, we refer to [18, 28]. Recently, by using the spectral equivalent properties similar to [4], Grief and Schötzau [3] construct the block-diagonal preconditioner: 𝑘=𝐴𝑘2𝑀+𝐵𝑇𝐿1𝐵00𝐿,(1.6) where 𝐿𝐑𝑚×𝑚 is the discrete Laplace operator introduced in [3], 𝑘2<1, and 𝑘 is a symmetric positive definite block-diagonal matrix. As 𝐿 is augmentation-free and Schur complement-free, this approach overcomes the difficulty in forming the Schur complement in general. However, the computational work of 𝐵𝑇𝐿1𝐵 may be too large. Using the fact that the matrices 𝐴+𝐵𝑇𝐿1𝐵 and 𝐴+𝑀 are spectrally equivalent, [3] considers the following preconditioner: 𝑘=𝐴+1𝑘2𝑀00𝐿,(1.7) and shows that the eigenvalues of the preconditioned matrix are tightly clustered.

Based on the work of Grief and Schötzau [3], [2] gives block-triangular Schur complement-free preconditioners for the linear system (1.2). And it is shown that all eigenvalues of the proposed block-triangular preconditioning matrices are more tightly clustered. Compared with the restriction 𝑘2<1 in [3], [2] considers the general case 𝑘2𝐑+. Furthermore, [15] provides block-triangular preconditioners when 𝑘2=0 with two variable relaxation parameters.

Based on the previous work [2, 3, 15], mentioned above, in this paper we are devoted to give new preconditioners with two scaling parameters. The new block triangular preconditioners in the general case 𝑘2𝐑+ contain the preconditioners discussed in [2]. Theoretical analysis shows that all eigenvalues of the preconditioned matrices are tightly clustered. Numerical experiments demonstrate efficiency of the new method and show that preconditioner 𝑘,𝜂,𝜀 is more efficient than 𝑘,𝑡.

The remainder of the paper is as follows. In Section 2, we establish new block-triangular preconditioners for the linear systems (1.2) in the general case 𝑘2𝐑+, and then the corresponding spectral analysis is presented. In Section 3, we provide numerical examples to examine our analysis. Finally, some conclusions are drawn in Section 4.

2. New Block-Triangular Preconditioners for Any 𝑘2

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 𝑘2𝐑+.

Grief and Schötzau [3] provide the block-diagonal Schur complement-free preconditioner 𝑘 as in (1.6). Using the fact that the matrices 𝐴+𝐵𝑇𝐿1𝐵 and 𝐴+𝑀 are spectrally equivalent, the argumentation-free and Schur complement-free preconditioner 𝑘 is defined in (1.7). Spectral analysis shows that the eigenvalues of the preconditioned saddle-point matrices 𝑘1𝒜 and 𝑘1𝒜 are strongly clustered when 𝑘2 is small.

Reference [2] provides the block-triangular Schur complement-free preconditioners for the linear system (1.2). In particular, they considered preconditioning matrices for the general case 𝑘2𝐑+ with 𝑘,𝑡=𝐴𝑘2𝑀+𝑡𝐵𝑇𝐿1𝐵𝑡𝐵𝑇01𝑡𝑡𝐿,𝑘,𝑡=𝐴+𝑡𝑘2𝑀𝑡𝐵𝑇01𝑡𝑡𝐿,(2.1) where 1𝑡>𝑘2.

For 𝑘2=0, [15] provides the block-triangular preconditioner for linear system (1.2): 𝒫𝜂,𝜀=𝐴+𝜂𝐵𝑇𝐿1𝐵(1𝜂𝜀)𝐵𝑇0𝜀𝐿,(2.2) where 𝜂>0 and 𝜀>0.

Based on the works in [2, 3, 15], we provide the following new block-triangular Schur complement-free preconditioners 𝑘,𝜂,𝜀 and 𝑘,𝜂,𝜀: 𝑘,𝜂,𝜀=𝐴𝑘2𝑀+𝜂𝐵𝑇𝐿1𝐵(1𝜂𝜀)𝐵𝑇0𝜀𝐿,(2.3)𝑘,𝜂,𝜀=𝐴+𝜂𝑘2𝑀(1𝜂𝜀)𝐵𝑇0𝜀𝐿,(2.4) where 𝜂>𝑘2 and 𝜀0 are scaling parameters. It is interesting to note that when parameters 𝜂=𝑡 and 𝜀=(1𝑡)/𝑡, 𝑘,𝜂,𝜀 and 𝑘,𝜂,𝜀 apparently reduce to 𝑘,𝑡 and 𝑘,𝑡, respectively. We also see that when 𝑘2=0, the preconditioner 𝑘,𝜂,𝜀 in (2.3) (𝜀0) is different from 𝒫𝜂,𝜀 in (2.2) (𝜀>0).

We stress that 𝑘,𝜂,𝜀 is not the preconditioner we eventually use in actual computation. It is only introduced to lay theoretical basis and motivation for the preconditioner 𝑘,𝜂,𝜀 in (2.4), which we will use in practice. We note that the (1,1) block 𝐴𝑘2𝑀+𝜂𝐵𝑇𝐿1𝐵 in 𝑘,𝜂,𝜀 is symmetric positive definite for 𝑘 is sufficiently small [3]. But this is not true when 𝑘 is large enough. However, this situation may not appear in the actual preconditioner 𝑘,𝜂,𝜀. The (1,1) block 𝐴+(𝜂𝑘2)𝑀 in 𝑘,𝜂,𝜀 is always symmetric positive definite when 𝜂>𝑘2. In this paper, we will apply the BiCGSTAB with the preconditioner 𝑘,𝜂,𝜀 as an outer solver for the saddle-point system (1.2). Then, the overall computational cost of solution procedure relies on how to efficiently solve the linear systems 𝐴+𝜏𝑀(𝜏=𝜂𝑘2) and 𝐿, which are called by inner solvers. For the linear system 𝐿 arising from a standard scalar elliptic problem, many efficient solution methods exist. On the other hand, for solving the linear system 𝐴+𝜏𝑀, we refer to [6, 8, 9, 14], and some detailed numerical examples are provided in [29].

For the spectral analysis, we recall some results which are contained in the following lemma.

Lemma 2.1 (see [3]). The following relations hold: (i)𝑛=null(𝐴)null(𝐵); (ii)𝑀𝑢𝐴,𝑢𝐵=0forany𝑢𝐴null(𝐴)andany𝑢𝐵null(𝐵); (iii)𝐵𝑇𝐿1𝐵𝑢𝐴,𝑢𝐴=𝑀𝑢𝐴,𝑢𝐴forany𝑢𝐴null(𝐴).

Theorem 2.2. Let 𝒜 be the saddle-point matrix in (1.2). Then the matrix 1𝑘,𝜂,𝜀𝒜 has two distinct eigenvalues, which are given by 𝜆1=1,𝜆21=𝜀𝜂𝑘2,(2.5) with the algebraic multiplicities n and m, respectively.

Proof. Suppose that 𝜆 is an eigenvalue of 1𝑘,𝜂,𝜀𝒜, whose eigenvector is 𝑣𝑞. Then the corresponding eigenvalue problem is 𝐴𝑘2𝑀𝐵𝑇𝑣𝑞𝐵0=𝜆𝐴𝑘2𝑀+𝜂𝐵𝑇𝐿1𝐵(1𝜂𝜀)𝐵𝑇𝑣𝑞0𝜀𝐿.(2.6)
From the second row we can obtain 𝑞=(1/𝜆𝜀)𝐿1𝐵𝑣. By substituting it into the first row we have 𝜆(1𝜆)𝐴𝑘2𝑀1𝑣+𝜀𝐵+𝜆𝜂𝑇𝐿1𝐵𝑣=0.(2.7)
It is straightforward to see that any vector 𝑣𝑛 satisfies (2.7) with 𝜆=1, so 𝜆=1 is an eigenvalue of 1𝑘,𝜂,𝜀𝒜. By the similar technique of linear independence considerations from [3], we can demonstrate that the eigenvalue 𝜆=1 has algebraic multiplicity 𝑛.
Since there are 𝑚 linearly independent null vectors of 𝐴, by Lemma 2.1, 𝑣=𝑣𝐴+𝑣𝐵𝑣𝐴0,𝑣𝐴null(𝐴),𝑣𝐵null(𝐵).(2.8) By Lemma 2.1 (ii) and (iii), and using the inner product in (2.7) with 𝑣𝐴, we have 1(1𝜆)𝜀+𝜆𝜂𝑘2||𝑣𝐴||2𝑀=0.(2.9) Since 𝑣𝐴0, from (2.9) we can obtain that 𝜆=1/𝜀(𝜂𝑘2) is another eigenvalue of 1𝑘,𝜂,𝜀𝒜, and we claim that the eigenvalue 𝜆=1/𝜀(𝜂𝑘2) has algebraic multiplicity 𝑚.

Corollary 2.3. Let 1/𝜀=𝜂𝑘2. Then the corresponding preconditioned matrix 1𝑘,𝜂,𝜀𝒜 has only one eigenvalue 𝜆=1 of algebraic multiplicity 𝑛+𝑚.

Proof. From Theorem 2.2, we can easily obtain the corresponding conclusion.

Remark 2.4. From Theorem 2.2, we demonstrate that the preconditioned matrix 1𝑘,𝜂,𝜀𝒜 has precisely two distinct eigenvalues. Then if Krylov subspace methods are used to solve (1.2) with 𝑘,𝜂,𝜀 as a preconditioner, the iteration will require merely two steps if round-off errors are ignored [30]. And from Corollary 2.3, for any 𝜂, we can find a number 𝜀 which makes the preconditioned matrix 1𝑘,𝜂,𝜀𝒜 have only one eigenvalue. Therefore, we can demonstrate that our preconditioners are more efficient than the block-triangular preconditioner proposed in [2].

Remark 2.5. From (2.3) we know that if 𝜂𝜀=1, the new preconditioner reduces to the diagonal preconditioner 𝑘,𝜂: 𝑘,𝜂=𝐴𝑘2𝑀+𝜂𝐵𝑇𝐿101𝐵0𝜂𝐿.(2.10) Then we can use MINRES to solve the linear system (1.2).

Theorem 2.6. Let 𝒜 be the saddle-point matrix in (1.2). Then 𝜆1=1,𝜆21=𝜀𝜂𝑘2(2.11) are the eigenvalues of 1𝑘,𝜂,𝜀𝒜, having algebraic multiplicity m. The rest of the eigenvalues satisfy 𝛼𝑘2𝛼+𝜂𝑘2𝜆<1,(2.12) where 𝛼 is defined as in (1.4).

Proof. Suppose that 𝜆 is an eigenvalue of 1𝑘,𝜂,𝜀𝒜, whose eigenvector is 𝑣𝑞. Then the corresponding eigenvalue problem is 𝐴𝑘2𝑀𝐵𝑇𝑣𝑞𝐵0=𝜆𝐴+𝜂𝑘2𝑀(1𝜂𝜀)𝐵𝑇𝑣𝑞0𝜀𝐿.(2.13) From the second row we can obtain 𝑞=(1/𝜀𝜆)(𝐿1𝐵𝑣). By substituting it into the first row we have 𝜆𝐴𝑘2𝑀1𝑣+𝜀𝐵𝑇𝐿1𝐵𝑣=𝜆2𝐴+𝜂𝑘2𝑀𝜆𝑣+𝜀(1𝜂𝜀)𝐵𝑇𝐿1𝐵𝑣.(2.14) Consider the 𝑚 linearly independent null vectors of 𝐴, by Lemma 2.1 (i), 𝑣=𝑣𝐴+𝑣𝐵𝑣𝐴0,(2.15) where 𝑣𝐴 null(A) and 𝑣𝐵 null(B). By Lemma 2.1 (ii) and (iii), and taking the inner product in (2.14) with 𝑣𝐴, we obtain 1(1𝜆)𝜀+𝜆𝜂𝑘2||𝑣𝐴||2𝑀=0.(2.16) Since |𝑣𝐴|𝑀0, 𝜆1=1 and 𝜆2=1/(𝜀(𝜂𝑘2)) are two eigenvalues of 1𝑘,𝜂,𝜀𝒜 and by the similar technique of linear independence considerations from [3], we claim that each eigenvalue has algebraic multiplicity 𝑚.
For the rest of eigenvectors we have 𝑣𝐵0. Noting that 𝐵𝑇𝐿1𝐵𝑣𝐴,𝑣𝐵=𝐿1𝐵𝑣𝐴,𝐵𝑣𝐵=0,(2.17) by Lemma 2.1 (ii) and by taking the inner product in (2.14) with 𝑣𝐵 and using (2.17), we obtain ||𝑣(1𝜆)𝐵||2𝐴=𝜆𝜂𝑘2+𝑘2||𝑣𝐵||2𝑀.(2.18) It is impossible to have 𝜆=1, since (2.18) leads to |𝑣𝐵|𝑀=0, which contradicts with 𝑣𝐵0. We cannot have 𝜆>1, since the left-hand side is negative but the right-hand side is positive (because we assume 𝜂>𝑘2). Thus, we claim that 𝜆<1.
From (1.4), we recall that for any 𝑢null(𝐵), |𝑢|2𝐴𝛼|𝑢|2𝑀 with 𝛼=𝛼/(1𝛼). Applying this to (2.18), we have (𝜆(𝜂𝑘2)+𝑘2)/(1𝜆)𝛼. Since 𝜂>𝑘2>0, we have 𝛼+𝜂𝑘2>0 and 𝛼𝑘2𝛼+𝜂𝑘2𝜆<1.(2.19)

Corollary 2.7. Let 1/𝜀=𝜂𝑘2. Then the corresponding preconditioned matrix 𝑘,𝜂,𝜀𝒜 has only one eigenvalue 𝜆=1 with algebraic multiplicity 2m. The remaining eigenvalues satisfy (2.12).

Remark 2.8. From (2.4) we know that when 𝜂𝜀=1, the new preconditioner reduces to the diagonal preconditioner 𝑘,𝜂: 𝑘,𝜂=𝐴+𝜂𝑘201𝑀0𝜂𝐿.(2.20) Then we can use MINRES to solve the linear system (1.2).

Remark 2.9. From the proof of Theorem 2.6, we easily see that the new preconditioner 𝑘,𝜂,𝜀 is also efficient for 𝑘2=0. Then from (2.12), we conclude that if 1/𝜀=𝜂𝑘2 and 𝑘2=0, then the closer 𝜂 is to 0 and the closer (𝛼𝑘2)/(𝛼+𝜂𝑘2)=𝛼/(𝛼+𝜂) is to 1; that is, the preconditioned matrix has more tightly clustered eigenvalues. For a general case of 𝑘2+, we can only obtain similar results when 𝛼>𝑘2. The following numerical experiments show that the closer 𝜂 is to 𝑘2, the less iteration counts we have used for a fixed 𝑘2+. However, choosing 𝜂𝑘2 too small may result in too large 𝜀, then result in ill-conditioning of 𝑘,𝜂,𝜀. So we choose 𝜂𝑘2 to be moderate size in practice.

3. Numerical Experiments

The test problem is a two-dimensional time-harmonic Maxwell equations in mixed form (1.1) in a square domain Ω=(0<𝑥<1;0<𝑦<1). We set the right-hand side function so that the exact solution is given by 𝑢𝑢(𝑥,𝑦)=1𝑢(𝑥,𝑦)2(=𝑥,𝑦)𝑦(1𝑦)𝑥(1𝑥)(3.1) and 𝑝0.

We consider five uniformly refined meshes, which are constructed by subsequently splitting each triangle into four triangles by joining the midpoints of the edges of the triangle. Two of five mesh grids are depicted in Figures 1 and 2. The lowest order elements are used to discretize equations. The matrix sizes on different meshes are given in Table 1.

tab1
Table 1: Values of matrix size of the linear system for five meshes.
617390.fig.001
Figure 1: 8×8 mesh.
617390.fig.002
Figure 2: 16×16 mesh.

Our numerical experiments were performed using MATLAB. The machine is a PC-Intel (R), Pentium(R)Dual CPU E2200 2.20 GHz, 1.00 G of RAM. The purpose of our experiments is to investigate the convergence behavior of preconditioned BiCGSTAB by choosing different parameters 𝜂 and and 𝜀 in the preconditioner 𝑘,𝜂,𝜀. Thus, we apply exact inner solver, and the outer iteration is used as a zero initial guess and stopped when 𝑟(𝑘)(=𝑏𝒜𝑥(𝑘))2/𝑟(0)(=𝑏)25×1010.

From Theorem 2.6 and Corollary 2.7 we know that the preconditioned matrix 1𝑘,𝜂,𝜀𝒜 has one eigenvalue 𝜆=1, and the remaining eigenvalues are satisfying (2.12). Figure 3 depicts the eigenvalues of the preconditioned matrix 1𝑘,𝜂,𝜀𝒜 with 𝑘2=2, where 𝜂 and 𝜀 satisfy 1/𝜀=𝜂𝑘2, and 𝑚+𝑛=961. From it we observe that the eigenvalue distribution of preconditioned matrix 1𝑘,𝜂,𝜀𝒜 with 𝜂=𝑘2+0.1=2.1 denoted by solid line is more tightly clustered than with 𝜂=𝑘2+30=32 denoted by dotted line. From Remark 2.9 we know that the new preconditioner 𝑘,𝜂,𝜀 is also efficient for 𝑘2=0. Figures 4, 5, 6, and 7 show the eigenvalue distribution of the preconditioned matrix 1𝑘,𝜂,𝜀𝒜 for different 𝜂 with 𝑘2=0, where 𝜂 and 𝜀 satisfy 1/𝜀=𝜂𝑘2, and 𝑚+𝑛=961. From Figures 47 we know that the closer the parameter 𝜂 is to 0, the more tightly clustered the eigenvalues of the preconditioned matrix will be.

617390.fig.003
Figure 3: The eigenvalue distribution of the preconditioned matrix 1𝑘,𝜂,𝜀𝒜 with 𝑘2=2, 𝜂=2.1 and 𝜂=32, respectively, and 𝑚+𝑛=961. The case 𝜂=2.1 is indicated by the solid line while the case 𝜂=32 is indicated by the dotted line.
617390.fig.004
Figure 4: The eigenvalue distribution of the preconditioned matrix 1𝑘,𝜂,𝜀𝒜 with 𝑘2=0, 𝜂=5 and 𝑚+𝑛=961.
617390.fig.005
Figure 5: The eigenvalue distribution of the preconditioned matrix 1𝑘,𝜂,𝜀𝒜 with 𝑘2=0, 𝜂=0.1 and 𝑚+𝑛=961.
617390.fig.006
Figure 6: The eigenvalue distribution of the preconditioned matrix 1𝑘,𝜂,𝜀𝒜 with 𝑘2=0, 𝜂=0.01 and 𝑚+𝑛=961.
617390.fig.007
Figure 7: The eigenvalue distribution of the preconditioned matrix 1𝑘,𝜂,𝜀𝒜 with 𝑘2=0, 𝜂=0.001 and 𝑚+𝑛=961.

Table 2 shows the outer iteration counts for different 𝑘2 and 𝜂, applying BiCGSTAB with the block-triangular preconditioner, where 𝜂 and 𝜀 satisfy 1/𝜀=𝜂𝑘2, and 𝑚+𝑛=16129. The iteration counts are denoted by Iter. We observe that for a fixed 𝑘2, the closer 𝜂 is to 𝑘2, the less iteration counts are produced. For comparison, we also give the outer iteration counts for 𝜂=𝑡=1+(𝑘2+1+𝑘4)/2. We refer the definition of 𝑡 to [2]. It shows that the preconditioner 𝑘,𝜂,𝜀 is more efficient than 𝑘,𝑡.

tab2
Table 2: Iteration counts for different 𝑘2 and 𝜂, using BiCGSTAB with the preconditioner 𝑀𝑘,𝜂,𝜀, and 𝑚+𝑛=16129. The iteration was stopped once 𝑟(𝑘)/𝑟(0)5×1010.

Tables 3 and 4 show the outer iteration numbers for different meshes, applying BiCGSTAB with the preconditioner 𝑘,𝜂,𝜀, where 𝜂 are set to be 𝜂𝑘2=0.1 and 𝜂𝑘2=6, and 1/𝜀=𝜂𝑘2. We observe that the outer iteration numbers of the preconditioned BiCGSTAB are hardly sensitive to the changes in the mesh size.

tab3
Table 3: Iteration counts for different meshes, using BiCGSTAB with the preconditioner 𝑀𝑘,𝜂,𝜀 satisfying 𝜂𝑘2=0.1 and 1/𝜀=𝜂𝑘2. The iteration was stopped once 𝑟(𝑘)/𝑟(0)5×1010.
tab4
Table 4: Iteration counts for different meshes, using BiCGSTAB with the preconditioner 𝑀𝑘,𝜂,𝜀 satisfying 𝜂𝑘2=6 and 1/𝜀=𝜂𝑘2. The iteration was stopped once 𝑟(𝑘)/𝑟(0)5×1010.

4. Conclusions

We have investigated the use of new block-triangular preconditioners with two variable relaxation parameters for solving the mixed formulation of the time-harmonic Maxwell equations. Our results are extensions of the work in [2, 3, 15]. The preconditioned matrices are demonstrated to have clustering eigenvalues. We have shown experimentally that the outer iteration numbers of BiCGSTAB with the new preconditioner are hardly any sensitive to the changes in the mesh size.

Acknowledgments

The authors thank the anonymous referees for their valuable comments and suggestions which lead to an improved presentation of this paper. This work is supported by the Chinese National Science Foundation Project (11161014), the Science and Technology Development Foundation of Guangxi (Grant no. 0731018), and Innovation Project of Guangxi Graduate Education (Grant no. ZYC0430).

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