Y. P. ZENG ET AL.

254

0,10 ,or1,00

.

When the parameters satisfy 11

, we can obtain

the following corollary from Theorem 2.3.

Corollary 2.4. Let 11

. Then the matrix 1

A

has one eigenvalue which given by λ = 1 with algebraic

multiplicity n + r. The remaining m − r eigenvalues lie in

the interval (0, 1).

3. Numerical Experiments

We consider the following finite element discretization

of the time-harmonic Maxwell equations (k2 = 0) [5,8].

The following two-dimensional Model problem is con-

sidered: find u and p that satisfy

in

0in

0on

0on

upf

u

u

p

n

(7)

Here is a simply connected polyhedron do-

main with a connected boundary ∂Ω, and ~n denotes the

outward unit normal on ∂Ω. The datum

2

R

is a given

source (not necessarily divergence free). Using the low-

est order N’ed’elec elements of first kind [9,10] for the

approximation of the vector field and standard nodal

elements for multiplier yields the following saddle-point

linear system

.

0

0

Tug

FB

x

p

B

b

Experiments were done in a square domain (0 ≤ x ≤ 1;

0 ≤ y ≤ 1). And we set the right-hand side function so

that the exact solution is given by

,1,1

T

uxyyyxx .

In our numerical experiments the matrix W in the

augmentation block preconditioner is taken as W = I.

We consider three meshes with different values of n

and m in Table 1. Table 2 shows iteration counts for

different η and meshes, applying BiCGSTAB of

block-triangular preconditioner, and 11

. We ob-

Table 1. Values of n and m and size of the linear systems for

three meshes.

h n m n + m

1

8 176 49 225

1

16 225 736 961

1

32 961 3008 3969

Table 2. Iteration counts for different and meshes, using

BiCGSTAB for solving the saddle point system with precon-

ditioner

, the iteration was stopped once

() 013k

rr

()10

.

h 0.1 0.5 2 4 6

1

8

h

3 3 1 1 0.5

1

16

h3 2 2 1 1

1

32

h1 0.5 0.5 0.5 0.5

serve that for a fixed mesh, When η = 6, We spent the

least iteration counts. In particulary, when η > 2, the it-

eration we spent are less than η = 2 (corresponding to M2

[1]). It shows that preconditioner

is more effi-

cient than t

and ˆt

.

4. Acknowledgements

The authors thanks the Chinese National Science Foun-

dation Project (11161014), the Science and Technology

Development Foundation of Guangxi (Grant No. 0731

018), Innovation Project of Guangxi Graduate Education

(Grant No. ZYC0430).

5. References

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Triangular Preconditioners for Saddle Point Linear Sys-

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