Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 245051, 16 pages
doi:10.1155/2012/245051
Research Article
Finite Element Preconditioning on Spectral Element
Discretizations for Coupled Elliptic Equations
JongKyum Kwon, Soorok Ryu, Philsu Kim, and Sang Dong Kim
Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea
Correspondence should be addressed to Sang Dong Kim, skim@knu.ac.kr
Received 8 August 2011; Accepted 2 November 2011
Academic Editor: Massimiliano Ferronato
Copyright q 2012 JongKyum Kwon et al. 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.
N 2 are shown both analytically and numerically by the
−12 A
The uniform bounds on eigenvalues of B
h
−1
2 for the Legendre spectral element system A
N 2 u f which is
P1 finite element preconditioner B
h
arisen from a coupled elliptic system occurred by an optimal control problem. The finite element
preconditioner is corresponding to a leading part of the coupled elliptic system.
1. Introduction
Optimal control problems constrained by partial differential equations can be reduced to
a system of coupled partial differential equations by Lagrange multiplier method 1. In
particular, the needs for accurate and efficient numerical methods for these problems have
been important subjects. Many works are reported for solving coupled partial differential
equations by finite element/difference methods; or finite element least-squares methods 2–
5, etc.. But, there are a few literature for examples, 6, 7 on coupled partial differential
equations using the spectral element methods SEM despite of its popularity and accuracy
see, e.g., 8.
One of the goals in this paper is to investigate a finite element preconditioner for the
SEM discretizations. The induced nonsymmetric linear systems by the SEM discretizations
from such coupled elliptic partial differential equations have the condition numbers which
are getting larger incredibly not only as the number of elements and degrees of polynomials
increases but also as the penalty parameter δ decreases see 5 and Section 4. Hence, an
efficient preconditioner is necessary to improve the convergence of a numerical method
whose number of iterations depends on the distributions of eigenvalues see 9–12.
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Particularly, the lower-order finite element/difference preconditioning methods for spectral
collocation/element methods have been reported 9, 10, 13–17, etc..
The target-coupled elliptic type equations are as follows:
1
v 0 in Ω,
δ
−Δv − ∇ · vb v − u −
u in Ω,
−Δu b · ∇u u uv0
1.1
on ∂Ω,
which is the result of Lagrange multiplier rule applied to a L2 optimal control problem subject
to an elliptic equation see 1. Applying the P1 finite element preconditioner to our coupled
elliptic system discretized by SEM using LGL Legendre-Gauss-Lobatto nodes, we show
that the preconditioned linear systems have uniformly bounded eigenvalues with respect to
elements and degrees.
The field of values arguments will be used instead of analyzing eigenvalues directly
because the matrix representation of the target operator, even with zero convection term,
is not symmetric. We will show that the real parts of eigenvalues are positive, uniformly
bounded away from zero, and the absolute values of eigenvalues are uniformly bounded
whose bounds are only dependent on the penalty parameter δ in 1.1 and the constant vector
b in 1.1. Because of this result, one may apply a lower-order finite element preconditioner
to a real optimal control problem subject to Stokes equations which requires an elliptic type
solver.
This paper is organized as follows: in Section 2, we introduce some preliminaries and
notations. The norm equivalences of interpolation operators are reviewed to show the norm
equivalence of an interpolation operator using vector basis. The preconditioning results are
presented theoretically and numerically in Section 3 and Section 4, respectively. Finally, we
add the concluding remarks in Section 5.
2. Preliminaries
2.1. Coupled Elliptic System
Because we are going to deal with a coupled elliptic system, the vector Laplacian, gradient
and divergence operators for a vector function u u, vT , where T denotes the transpose,
are defined by
Δu :
uxx uyy
,
vxx vyy
∇u :
ux vx
,
uy vy
∇ · u :
u x uy
.
vx vy
2.1
With the usual L2 inner product ·, · and its norm · , for vector functions u : u, vT and
w : w, zT , define
u, w : u, w v, z,
2
u : u2 v2
2.2
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3
and, for matrix functions U and V , define
U, V 4
uk , vk ,
4
U :
uk 2 ,
2
k1
k1
u1 u2
v1 v2
where U , V .
u3 u4
v3 v4
2.3
We use the standard Sobolev spaces like H 1 Ω and H01 Ω on a given domain Ω with
the usual Sobolev seminorm | · |1 and norm · 1 . The main content of this paper is to provide
an efficient low-order preconditioner for the system 1.1.
Multiplying the second equation by 1/δ, the system 1.1 can be expressed as
T
Au : −AΔu B b · ∇u A Cu f
2.4
in Ω,
where
⎤
⎡
1 0
A ⎣ 1 ⎦,
0
δ
⎡
⎤
1 0
B⎣
1 ⎦,
0 −
δ
⎡
⎤
1
0
⎢
⎥
C ⎣ 1 δ ⎦,
0
−
δ
⎤
0
f ⎣ 1 ⎦,
− u
δ
⎡
2.5
with the zero boundary condition u 0 on ∂Ω. Let B be another decoupled uniformly elliptic
operator such that
Bu : −AΔu Au
in Ω
2.6
with the zero boundary condition.
2.2. LGL Nodes, Weights, and Function Spaces
N
Let {ηk }N
k0 and {ωk }k0 be the reference LGL nodes and its corresponding LGL weights in
I −1, 1, respectively, arranged by −1 : η0 < η1 < · · · < ηN−1 < ηN : 1. We use {tj }Ej0 as
the set of knots in the interval I such that −1 : t0 < t1 < · · · < tE−1 < tE : 1. Here E denotes
the number of subintervals of I. Denote N by the degree of a polynomial on each subinterval
by the set of kth-LGL nodes ξj,k in each subinterval Ij
Ij : tj−1 , tj and G : {ξj,k }E,N
j 1,k 0
j 1, . . . , E arranged by
ξj,0 : tj−1 < ξj,1 < · · · < ξj,N−1 < tj : ξj,N ,
2.7
where
ξj,k hj
1
ηk tj−1 tj ,
2
2
hj tj − tj−1 ,
2.8
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are given by
and the corresponding LGL weights {ρj,k }E,N
j1,k0
ρj,k
⎧
hj
1
hj
⎪
⎪
⎪ ωN ω0 ,
⎪
⎪
2
2
⎪
⎨
: ρj−1,N,
⎪
⎪
⎪
⎪
⎪
h
⎪
⎩ j ωk ,
2
k N, j 1, . . . , E − 1,
k 0, j 2, . . . , E,
2.9
otherwise.
Let Pk be the space of all polynomials defined on I whose degrees are less than or equal to k.
N
The Lagrange basis for PN on I is given by {φi t} satisfying
i0
φi ηj δij
2.10
for i, j 0, 1, . . . , N,
where δij denotes the Kronecker delta function.
h
We define PN
as the subspace of continuous functions whose basis {φμ }N
1
is
μ0
piecewise continuous Lagrange polynomials of degree N on Ij with respect to G and define
h
VN
as the space of all piecewise Lagrange linear functions {ψμ }N
1
with respect to G. Note
μ0
0
h
that these basis functions have a proper support. See 9, 18 for detail. Let PN,h
: H01 I ∩ PN
0
0
h
and VN,h
: H01 I ∩ VN
. With the notation V × V : {u, vT | u, v ∈ V}, define PN,h
and V0N,h
0
0
0
0
0
as subspaces of PN,h × PN,h and VN,h × VN,h , respectively. The basis functions for PN,h
and
0
VN,h are given respectively by
T
φp t, 0 ,
φ t : T
p
0, φp−N t ,
for p ≤ N,
for p > N,
T
ψp t, 0 ,
ψ t : T
p
0, ψP −N t ,
for p ≤ N,
for p > N,
2.11
0
0
0
: PN,h
⊗ PN,h
and
where p 1, 2, . . . , 2N. For two dimensional 2D case, let PN,h
0
0
0
VN,h : VN,h ⊗ VN,h be tensor product function spaces of one-dimensional function spaces
0
0
0
0
and let P0N,h and V0N,h be subspaces of PN,h
× PN,h
and VN,h
× VN,h
,
:
respectively. Now let us order the interior LGL points in Ω by horizontal lines as {Ξμ }N
μ
1
2
, where μ
μ
Nν−1 for μ, ν 1, 2, . . . , N. Accordingly, the basis functions for
{ξμ , ξν }N,N
μ1,ν1
P0N,h and V0N,h are also arranged as the same way. Then, with the notations φμ x, y :
0
and V0N,h are given,
φμ xφν y and ψμ x, y : ψμ xψν y, the basis functions of PN,h
respectively, by
⎧ ⎨ φp x, y , 0 T ,
for p ≤ N2 ,
Φp x, y : ⎩ 0, φ 2 x, y
T , for p > N2 ,
p−N
⎧ T
for p ≤ N2 ,
⎨ ψp x, y , 0 ,
Ψp x, y : ⎩ 0, ψ 2 x, y T , for p > N2 ,
p−N
where p 1, . . . , 2N2 .
2.12
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5
2.3. Interpolation Operators
We denote CΩ as the set of continuous functions in Ω : I × I. Let IN 2 : CΩ → PN :
PN ⊗ PN be the usual reference interpolation operator such that IN 2 uηi , ηj uηi , ηj 0
h
for u ∈ CΩ see, e.g., 17. The global interpolation operator IN
2 : CΩ → PN,h 2
N
h
is given by IN
φμ
x, y, where uμ
uξμ , ξν . Hence, it follows that
2 ux, y μ
1 uμ
h
h
IN 2 uξμ , ξν uξμ , ξν for u ∈ CΩ. With this interpolation operator IN
2 , let us define
h
0
the vector interpolation operator IN 2 : CΩ × CΩ → PN,h such that, for u : u, vT ∈
CΩ × CΩ,
h h T
IhN 2 u ξμ , ξν : IN
u ξμ , ξν .
2 u ξμ , ξν , IN 2 v ξμ , ξν
2.13
x, y ψij x, y ψi xψj y, where i i N x, y φij x, y φi xφj y and Ψ
Let Φ
i
i
1j and i, j 0, . . . , N, be the basis of PN and VN , respectively. Let us denote MN 2 and
Mh2 by the mass matrices such that
,
, Φ
MN 2 i, j Φ
i
j
, Ψ
Mh2 i, j Ψ
i
j
2.14
and denote SN 2 and Sh2 by the stiffness matrices such that
, ∇Φ
,
SN 2 i, j ∇Φ
i
j
, ∇Ψ
,
Sh2 i, j ∇Ψ
i
j
2.15
where i, j 1, . . . , N 12 .
According to Theorems 5.4 and 5.5 in 17, there are two absolute positive constants c0
and c1 such that for any U u1 , . . . , uN
12 T ,
c0 MN 2 U, U ≤ Mh2 U, U ≤ c1 MN 2 U, U,
c0 SN 2 U, U ≤ Sh2 U, U ≤ c1 SN 2 U, U,
2.16
and for all u ∈ VN ,
c0 u ≤ IN 2 u ≤ c1 u,
c0 u1 ≤ IN 2 u1 ≤ c1 u1 .
2.17
h
The extension of 2.17 to the interpolation operator IN
2 leads to
h c0 u ≤ IN
2 u ≤ c1 u,
h c0 u1 ≤ IN
2 u ≤ c1 u1
1
2.18
0
for all u ∈ VN,h
, where the constants c0 and c1 are positive constants independent of E
and N see 18.
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Journal of Applied Mathematics
0
, there are positive constants c0 and c1 independent of E and N
Theorem 2.1. For all u ∈ VN,h
such that
c0 u ≤ IhN 2 u ≤ c1 u,
c0 u1 ≤ IhN 2 u ≤ c1 u1 .
2.19
1
h
Proof. By the definitions of the interpolation operator IN
2 and the norms, we have
h 2 h 2 h 2
IN 2 u IN 2 u IN 2 v ,
h 2 h 2 h 2
IN 2 u IN 2 u IN 2 v ,
1
1
1
2.20
which completes the proof because of 2.18.
3. Analysis on P1 Finite Element Preconditioner
The bilinear forms corresponding to 2.4 and 2.6 are given by
T α u, w A∇uT , ∇wT B b · ∇u , w A Cu, w f, w ,
3.1
β u, w A∇uT , ∇wT Au, w ,
3.2
where u : ux, y, vx, yT , w : wx, y, zx, yT and A, B, C are the same matrices in
x, yT . Note that the bilinear form β·, · in 3.2 is symmetric but the
2.4 and f 0, u
bilinear form α·, · in 3.1 is not symmetric. The following norm equivalence guarantees the
existence and uniqueness of the solution in H01 Ω × H01 Ω for the variational problem 3.1.
Proposition 3.1. For a real valued vector function ux, y u, vT ∈ H01 Ω × H01 Ω, we have
u ≤ α u, u ≤ 1 u2 .
1
1
δ
3.3
Proof. Since b is a constant vector, u, vT ∈ H01 Ω × H01 Ω, and u is a real valued vector
function, we have
T α u, u β u, u B b · ∇u , u Cu, u
1
1
β u, u b · ∇u, u − b · ∇v, v v, u − u, v
δ
δ
1
β u, u u21 v21 .
δ
Hence, 3.3 is proved because 0 < δ ≤ 1.
3.4
Journal of Applied Mathematics
7
Lemma 3.2. If u u, vT is a complex valued function in H01 Ω × H01 Ω, then we have the
following estimates:
Re α u, u β u, u ,
!
"
1
1 β u, u ,
α u, u ≤ 1 |b| 2
δ
where |b| 3.5
#
b12 b22 .
Proof. Let us decompose u and v in u as ux, y px, y iqx, y and vx, y rx, y isx, y, where p, q, r, and s are real functions and i2 −1. Then we have
$
b · ∇u, u i
Ω
1
i
− b · ∇u, u −
δ
δ
b1 qx p − px q b2 py q − qy p dΩ,
$
Ω
b1 sx r − rx s b2 ry s − sy r dΩ,
"
! $
1
2
sp − qr dΩ .
v, u − u, v i
δ
δ Ω
3.6
3.7
3.8
Hence, one may see that the real part of αu, u is βu, u and the pure imaginary part is the
sum of 3.6, 3.7, and 3.8. By Cauchy-Schwarz inequality, -inequality, and the range of
0 < δ ≤ 1, we have
1
1
b · ∇v, v v, u − u, v
δ
δ
1
1
≤ |b|∇uu |b|∇vv 2vu
δ
δ
!
" 1 1
1
≤ |b| ∇u2 u2 ∇v2 v2
u2 v2
2
δ
δ
!
"
1
1 ≤
β u, u .
|b| 2
δ
b · ∇u, u −
3.9
Hence 3.5 is proved.
Let σA and FA be the spectrum or set of eigenvalues and field of values of the
N 2 and B
h2 be the two dimensional stiffness matrices on
square matrix A, respectively. Let A
0
0
the spaces PN,h and VN,h induced by 3.1 and 3.2, respectively. Then, we have the
following.
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Journal of Applied Mathematics
2
Lemma 3.3. For U /
0 ∈ C2N , one has
−12 A
N 2 ⊂ W,
σ B
h
3.10
⎧
⎫
N 2 U, U
⎨ A
⎬
W :
| U
/0 .
⎩ B
⎭
h2 U, U
3.11
where
h2 is symmetric positive definite, there exists a unique positive definite square
Proof. Since B
1/2
2 of B
h2 . So, we have
root B
h
−1/2
N 2 U, U
N 2 U, U
−1/2
2
A
B
A
A
B
V,
V
N
h2
h2
,
1/2
1/2
V
V,
h2 U, U
2 U, B
2 U
B
B
h
h
1/2
U,
where V B
h2
3.12
−1
2
−1/2
1/2
for U /
0 ∈ C2N . Now let B
be a short-hand notation for B
. Therefore, from the
h2
h2
relation of spectrum and field of values see 19 or 11, it follows that
N2 σ B
N2 B
−1/2
N2 B
−1/2
−12 A
−1/2
−1/2
A
A
σ B
⊂F B
W,
h
h2
h2
h2
h2
3.13
which completes the proof.
The following theorem shows the uniform bounds of eigenvalues which is independent of both N and E for our preconditioned system
N 2 uN B
−12 A
−12 fN .
B
h
h
3.14
2
Theorem 3.4. Let {λp }2N
be the set of eigenvalues of
p1
−12 A
N2 ,
B
h
3.15
Journal of Applied Mathematics
9
then, there are constants c0 , C0 , and Λδ independent of E and N, such that
0 < c0 < Re λp < C0 ,
λp ≤ Λδ ,
3.16
where Λδ C1 b 1/δ.
0
Proof. Let ux, y ∈ VN,h
be represented as ux, y polynomial interpolation can be written as
IhN 2 u
x, y 2N
2
p1
2N2
p1
up Ψp x, y. Then, its piecewise
up Φp x, y .
3.17
Let U u1 , . . . , u2N2 T . From the definitions of the bilinear forms, we have
N 2 U, U α Ih 2 u, Ih 2 u ,
A
N
N
h2 U, U β u, u .
B
3.18
This implies that
N 2 U, U
A
α IhN 2 u, IhN 2 u
.
wλ : β u, u
h2 U, U
B
3.19
By Theorem 2.1 and Lemma 3.2, the real part of wλ satisfies
β IhN 2 u, IhN 2 u
Rewλ ∼ 1,
β u, u
3.20
where the notation a ∼ b means the equivalence of two quantities a and b which does not
depend on E and N. Again, from Theorem 2.1 and Lemma 3.2, the absolute value of wλ
satisfies
α IhN 2 u, IhN 2 u
1 1/2|b| 1/δ β IhN 2 u, IhN 2 u
≤
|wλ | β u, u
β u, u
!
"
1
≤ C 1 |b| .
δ
3.21
−12 A
N2
Therefore, from Lemma 3.3, the real parts and the absolute values of eigenvalues of B
h
satisfy 3.16.
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Journal of Applied Mathematics
Remark 3.5. Let the one dimensional 1D bilinear forms for u, w ∈ H01 −1, 1 × H01 −1, 1 be
α1 u, w Au , w B bu , w A Cu, w ,
β1 u, w Au , w Au, w ,
3.22
3.23
N and B
h by the 1D stiffness matrices on the spaces P0 and V0 corresponding
and denote A
N,h
N,h
to the bilinear forms 3.22 and 3.23, respectively. Then one can easily get the same results
as Theorem 3.4 for 1D case. Since the proof is similar to 2D case, we omit the statements.
Now, for actual numerical computations, we need 2D stiffness and mass matrices
expressed as the tensor products of 1D matrices see 20 for details. For this, let us denote
1D spectral element matrices as
SN φμ t, φν t ,
RN φμ t, φν t ,
MN φμ t, φν t
3.24
Mh ψμ t, ψν t ,
3.25
and 1D finite element matrices
Sh ψμ t, ψν t ,
Rh ψμ t, ψν t ,
0
0
where {φμ }N
and {ψν }N
ν1 are the Lagrange basis of the spaces PN,h and VN,h , respectively.
μ1
0
For actual computations for SN , RN , and MN , the inner product ·, · on the space PN,h
will be computed using LGL quadrature rule at LGL nodes. Without any confusion, such
approximate matrices can be denoted by same notations in the next section. We note that the
approximate matrices SN , RN , and MN and the exact matrices SN , RN , and MN are equivalent,
respectively, because of the equivalence of LGL quadrature on the polynomial space we used.
For example, the mass matrix MN can be computed using the LGL weights {ρj,k } only.
4. Numerical Tests of Preconditioning
4.1. Matrix Representation
In this section we discuss effects of the proposed finite element preconditioning for the
spectral element discretizations to the coupled elliptic system 1.1. For this purpose, first
N and B
h corresponding to 3.22 and 3.23 using the
we set up one dimensional matrices A
matrices in 3.24. One may have
N
A
⎡
⎤
1
M
bR
M
S
N
N
N
N
⎢
⎥
δ
⎣
,
⎦
1
1
− MN
SN − bRN MN δ
δ
2N×2N
⎤
⎡
0
Sh Mh
h ⎣
⎦
B
.
1
0
Sh Mh δ
2N×2N
4.1
Journal of Applied Mathematics
11
N 2 and B
h2 can be
Let b b1 , b2 T be a constant vector in 1.1, then the 2D matrices A
expressed as
N2
A
⎡
⎤
1
2
2
2
2
M
R
M
S
N
N
N
⎢ N
⎥
δ
,
⎣
⎦
1
1
SN 2 − RN 2 MN 2 − MN 2
δ
δ
2N2 ×2N2
h2
B
⎤
⎡
Sh2 Mh2
0
⎦
,
⎣
1
0
Sh2 Mh2 δ
2N2 ×2N2
4.2
where
Sk2 Mk y ⊗ Sk x Sk y ⊗ Mk x,
Mk2 Mk y ⊗ Mk x
RN 2 b1 MN y ⊗ RN x b2 RN y ⊗ MN x.
k N or h,
4.3
4.2. Numerical Analysis on Eigenvalues
N 2 uN B
N 2 uN fN and the preconditioned linear system B
−12 A
−12 fN
The linear system A
h
h
will be compared in the sense of the distribution of eigenvalues. As proved in Section 3, it
N 2 are independent of the number of elements
−12 A
is shown that the behaviors of spectra of B
h
and degrees of polynomials.
One may also see the condition numbers of these discretized systems by varying the
N 2 are presented in Figure 1 for fixed δ 1
penalty parameter δ. The condition numbers of A
left and fixed E 3 right as increasing the degrees N of polynomials. It shows that such
condition numbers depend on N, E, and δ. In particular, the smaller δ is, the larger condition
number it yields.
N 2 for
−12 A
Figures 2 and 4 show the spectra of the resulting preconditioned operator B
h
−4
the polynomials of degrees N 4, 6, 8, 10 and E 4 when δ 1 and δ 10 , respectively.
−12 A
N 2 for E 4, 6, 8, 10 and N 4 for δ 1 and
Also, Figures 3 and 5 show the spectra of B
h
δ 10−4 , respectively. The same axis scales are presented for the same δ when b 1, 1T .
−12 A
N 2 are independent of N and E, but they
As proven in Theorem 3.4, the eigenvalues of B
h
depend still on the penalty parameter δ.
By choosing the convection coefficient b 10, 10T , in Figure 6, the distributions of
N 2 left and B
N 2 right are presented for penalty parameters δ 1, 10−3
−12 A
eigenvalues of A
h
to examine their dependence. In this figure, we see that the distributions of eigenvalues
N 2 are strongly dependent on δ. The real parts of
both real and imaginary part of A
such eigenvalues are increased in proportion to 1/δ. On the other hand, as predicted by
N 2 are positive and uniformly bounded
−12 A
Theorem 3.4, the real parts of the eigenvalues of B
h
away from 0. Moreover, the real parts are not dependent on δ and b see Figures 2–6, and
their moduli are uniformly bounded. The numerical results show that the imaginary parts
of the eigenvalues are bounded by some constants which are dependent on δ and b. These
phenomena support the theory proved in Theorem 3.4.
12
Journal of Applied Mathematics
×104
12
Condition number
Condition number
4000
3000
2000
1000
0
10
8
6
4
2
0
4
6
8
10
12
4
14
6
8
10
12
14
Degree of polynomial (N)
Degree of polynomial (N)
δ=1
δ = 0.1
δ = 0.01
E=1
E=2
E=3
a
b
N 2 when b 1, 1T .
Figure 1: The condition numbers of the matrix A
N = 4, E = 4
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
0
1
2
3
N = 6, E = 4
1.5
4
5
6
−1.5
0
1
2
a
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
0
1
2
5
6
3
c
4
5
6
N = 10, E = 4
1.5
1
−1.5
4
b
N = 8, E = 4
1.5
3
4
5
6
−1.5
0
1
2
3
d
N 2 for N 4, 6, 8, 10 and E 4 when δ 1, b 1, 1T .
−12 A
Figure 2: The eigenvalues of B
h
5. Concluding Remarks
An optimal control problem subject to an elliptic partial differential equation yields coupled
elliptic differential equations 1.1. Any kind of discretizations leads to a nonsymmetric linear
system which may require Krylov subspace methods to solve the system. In this paper, the
Journal of Applied Mathematics
13
N = 4, E = 4
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
N = 4, E = 6
1.5
0
1
2
3
4
5
6
−1.5
0
1
2
a
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
0
1
2
5
6
3
5
6
N = 4, E = 10
1.5
1
−1.5
4
b
N = 4, E = 8
1.5
3
4
5
6
−1.5
0
1
2
c
3
4
d
−12 A
N 2 for E 4, 6, 8, 10 and N 4 when δ 1, b 1, 1T .
Figure 3: The eigenvalues of B
h
N = 4, E = 4
15
10
10
5
5
0
0
−5
−5
−10
−10
−15
0
1
2
3
N = 6, E = 4
15
4
5
6
−15
0
1
2
a
10
5
5
0
0
−5
−5
−10
−10
0
1
2
3
c
5
6
4
5
6
N = 10, E = 4
15
10
−15
4
b
N = 8, E = 4
15
3
4
5
6
−15
0
1
2
3
d
N 2 for N 4, 6, 8, 10 and E 4 when δ 10−4 , b 1, 1T .
−12 A
Figure 4: The eigenvalues of B
h
14
Journal of Applied Mathematics
N = 4, E = 4
15
10
10
5
5
0
0
−5
−5
−10
−10
−15
0
1
2
3
N = 4, E = 6
15
4
5
6
−15
0
1
2
a
15
10
5
5
0
0
−5
−5
−10
−10
−15
1
2
5
6
5
6
N = 4, E = 10
10
0
4
b
N = 4, E = 8
15
3
3
4
5
6
−15
0
1
2
c
3
4
d
−12 A
N 2 for E 4, 6, 8, 10 and N 4 when δ 10−4 , b 1, 1T .
Figure 5: The eigenvalues of B
h
δ = 1, [b1 , b2 ] = [10, 10]
δ = 1, [b1 , b2 ] = [10, 10]
2
2
1
1
0
0
−1
−1
−2
−2
0
5
10
15
0
20
1
2
3
4
5
6
b
a
δ = 0.0001, [b1 , b2 ] = [10, 10]
δ = 0.0001, [b1 , b2 ] = [10, 10]
40
40
20
20
0
0
−20
−20
−40
−40
0
0.5
1
1.5
c
2
×104
0
1
2
3
4
5
6
d
N 2 left and B
−12 A
N 2 right for δ 1 top and δ 10−3 bottom when
Figure 6: The eigenvalues of A
h
T
b 10, 10 and N 12, E 4.
Journal of Applied Mathematics
15
spectral element discretization is chosen because it is very accurate and popular, but the
resulting linear systems have large condition numbers. This situation now becomes one
of disadvantages if one aims at a fast and efficient numerical simulation for an optimal
control problem subject to even a simple elliptic differential equation. To overcome such
a disadvantage, the lower-order finite element preconditioner is proposed so that the
preconditioned linear system has uniformly bounded condition numbers independent of the
degrees of polynomials and the mesh sizes. One may also take various degrees of polynomials
on subintervals with different mesh sizes. In this case, similar results can be obtained without
any difficulties. This kind of finite element preconditioner may be used for an optimal control
problem subject to Stokes flow see, e.g., 13.
Acknowledgments
The authors would like to thank Professor. Gunzburg for his kind advice to improve this
paper. This work was supported by KRF under contract no. C00094.
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