355

THE APPLICATIONS OF CG AND PCG ON

ELLIPTIC PARTIAL DIFFERENTIAL

EQUATIONS SOLVED BY FINITE ELEMENT

METHOD

Dr. Omar Ali Aleyan

Abstract

We apply the CG and PCG methods to the linear system

.bHx which was derived from elliptic partial differential

equation by using finite element method in order to get the

number of iteration and residual. Two examples will be used

for implications.

Key words

CG and PCG methods, Elliptic partial differential

equations, finite element methods.

Introduction

Let be a bounded domain in ,R d with boundary .

ـــــــــــــــــــــــــــــــــــــــ Mathematics Department, Faculty Of Education, University Of AL Asmarya.

THE APPLICATIONS OF CG AND PCG ON ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

سمرية مجلة الجامعة الأ435

It is assumed that f and g are continuous on . So a unique

solution exists

y)f(x, u For andy)(x,

u(x,y)=g(x,y) in

.dycb,xa:y)(x,

for the basis of the finite element approximation of (1).

First the test function v is picked where v satisfies the

boundary condition 0v on . Multiply the first equation by

v, and integrate the equation over , and the Green's formula,

is used

dx, vu ds v

n

udx u v

A finite element discretization of (1) is based on the

weak formulation: seek d1

0 )](H[Vu such that

Vv v)(f,p)b(v,v)a(u,

where dx vu v)a(u,

, dx vfv)(f,

.

the approximate solution VVu hh satisfies

hh Vv v)(f,v),a(u

If

N

1i

iih )x(uu and substitutes them in the equations

Dr. Omar Ali Aleyan

31السنة العدد 434

N1,...,k )(f,),a(u kkh

It results in

N1,....,k )(f,),.a(u k

N

1i

kii

It can be rewritten in the following form

bHx .

Here nnRH is the symmetric matrix ,Rb n and for the

solution nRx is obtained

In this paper we will use conjugate gradient method

and preconditioned conjugate gradient methods to solve the

system of linear equations.

bHx

[1] Conjugate gradient method. (CG)

We will use the conjugate gradient method to solve the

system of linear algebraic equations

bHx

We choose an initial approximation0x , put

00 Hxbr , 00 rp

and compute

j

T

j

j

T

j

jHpp

rra ,

THE APPLICATIONS OF CG AND PCG ON ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

سمرية مجلة الجامعة الأ435

jjj1j paxx

,Hparr jjj1j

j

T

j

1j

T

1j

jrr

rrb

,

.pbrp jj1j1j

[2] Preconditioned Conjugate gradient method. (PCG)

Let us denote B as the preconditioner of the symmetric

positive definite matrix where the matrix B is closed to the

matrix H of the system.

If we put

H=B-E (1.4)

Then we can, e.g., require that the norm HBE to be

small.

It is well known that for every symmetric positive

definite matrix B, there exists just one symmetric positive

definite matrix 2

1

B

(the square root of 1B ) such that

.BBB 2

1

2

1

1

Thus, put

bBb 2

1

and rewrite the system (1) as

Dr. Omar Ali Aleyan

31السنة العدد 436

,bxH

where

,HBBH 2

1

2

1

.xBx 2

1

We now can solve the system (1.6) using the conjugate

gradient method (1.2), (1.3) and the solution x computed from

Notes that, according to (1.4) and (1.7),

,EIEBBIH 2

1

2

1

where we put 2

1

2

1

EBBE

.

After substituting b and x ,H from the formulae (1.5),

(1.7) and (1.8), the algorithm (1.2) and (1.3) for solving the

system (1.6) can be rewritten as follows. Choose the initial

approximation ,x 0 put

,rp ,xBHBBbBr 0002

1

2

1

2

1

2

1

0

and compute

,

pHBBp

rra

j2

1

2

1

T

j

j

T

j

j

,paxBxB jjj2

1

1j2

1

,pHBBarr j

2

1

2

1

jj1j

THE APPLICATIONS OF CG AND PCG ON ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

سمرية مجلة الجامعة الأ437

,rr

rrb

j

T

j

1j

T

1j

j

.pbrp jj1j1j

Substituting j

2

1

jj2

1

j pBp and rBr

we finally get the

preconditioned algorithm (instead of jj p and r we write

jj p and r ):

Choose the initial approximation ,x 0 put

,rBp ,Hxbr 0

1

000

and compute

,Hpp

rBra

j

T

j

j

1T

j

j

,paxx jjj1j

,Hparr jjj1j

,rBr

rBrb

j

1T

j

1j

1T

1j

j

.pbrBp jj1j

1

1j

How to choose starting and stopping points?

If you have a rough estimate of the value of x, use it as

the starting value0x . If not, set 0.x 0

When CG reaches the

minimum point, the residual becomes zero; we must stop

immediately when the residual is zero.

Dr. Omar Ali Aleyan

31السنة العدد 438

The preconditioners

The preconditioner of tridiagonal of the matrix H ( .BT)

The matrix H is preconditioned by the preconditioner

matrix tridiag(H)BT and HBT 1

TT

is the preconditioned

matrix.

[2]The preconditioner of incomplete LU factorization of the

matrix H )(BILU

The matrix H is preconditioned by the preconditioner

matrix ILUB where L is a lower triangular matrix and U is an

upper triangular matrix and HBT 1

ILUILU

is the preconditioned

matrix.

Results and Applications

We will use two examples for the conjugate gradient

method and preconditioned conjugate gradient method

Example 1

Consider the problem in the given domain with boundary

conditions:

boundaryin 0y)u(x,

1yx,0)y(x2Δu 22

Example 2

Consider the problem in the given domain with boundary

conditions

1yx,06y)y2-(xeΔu 3x

boundaryin 0y)u(x,

THE APPLICATIONS OF CG AND PCG ON ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

سمرية مجلة الجامعة الأ455

Figure 1.1 The shape of example 1 where n = 20

Figure 1.2 The final shape of matrix H of example 1

where n = 10

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

0

0.05

0.1

0.15

0 20 40 60 80 100 120

0

20

40

60

80

100

120

nz = 445

Dr. Omar Ali Aleyan

31السنة العدد 455

Figure 1.3 The shape of example 2 where n = 20

Figure 1.4 The final shape of matrix H of example 2

where n = 10

00.2

0.40.6

0.81

0

0.5

1-0.02

0

0.02

0.04

0.06

0.08

0.1

0 20 40 60 80 100 120

0

20

40

60

80

100

120

nz = 445

THE APPLICATIONS OF CG AND PCG ON ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

سمرية مجلة الجامعة الأ455

The results of numerical tests of the conjugate gradient

method and preconditioned conjugate gradient method for

example 1 and example 2 are given in the following tables:

Table 1 Iteration and residual of matrix H by the

conjugate gradient method

of example 1

h

8.6261e- 8.7947e- 9.2641e- 6.4411e- Residual CG

Iteration

Table 2 Iteration and residual of matrix H by the

preconditioned conjugate

gradient method of example 1

h

8.0969e- 8.8534e- 7.7129e- 5.4801e- Residual TT

Iteration

1.4968e- 1.5134e- 7.4100e- 3.2854e- Residual ILUT

Iteration

Table 3 Iteration and residual of matrix H by the

conjugate gradient method

of example 2

h

9.6951e- 9.7013e- 9.9616e- 7.4896e- Residual CG

Iteration

Table 4 Iteration and residual of matrix H by the

Dr. Omar Ali Aleyan

31السنة العدد 453

preconditioned conjugate

gradient method of example 2

h

7.7787e- 9.3371e- 7.3937e- 6.4889e- Residual TT

Iteration

1.2640e- 4.7331e- 2.9969e- 3.4696e- Residual ILUT

Iteration

Conclusion. It is shown in this paper that in the CG

method the number of iterations increases when the size of

matrix H increases. The convergent in the preconditioned

matrix is faster than in the non preconditioned one. The best

preconditioner is the incomplete preconditioner.

THE APPLICATIONS OF CG AND PCG ON ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

سمرية مجلة الجامعة الأ455

References

Aleyan O. A.: Applications of Finite Element Method to

Solve Elliptic Partial Differential Equations Using

Iterative Methods. An Academic, Intellectual, Cultural,

Comprehensive and Arbitrated Journal Issued by the

Faculty of Arts and Sciences in Zliten, Al-Mergeb. Issue

No. 21 2012 AS –SAYIF (December) .

Aleyan O. A.: On matrix splitting and application into

iterative methods for linear systems An Academic,

Intellectual, Cultural, Comprehensive and Arbitrated

Journal Issued by the Faculty of Arts and Sciences in

Zliten, Al-Merqab. Issue No. 18 2009 AS –SAYIF

(JUNE) 1377.

Berman, A. and Plemmons, R. Nonnegative matrices in

the mathematical sciences, Academic Press, New York,

Horn, R.A. and Johnson, C.R. Matrix analysis, Cambridge

University Press, Cambridge, 1986.

Ortega, J.M. and Rheinboldt, W.C. Iterative solution of

linear systems, Plenum Press, New York, 1988.

Miroslav Fiedler., Special matrices and their applications

in numerical mathematics, Kluwer, 1986.

Miroslav Fiedler and Vlastimil Ptak, On matrices with

non-positive off-diagonal elements and positive principal

minors, Czechosl. Math. Journal (87)12,1962, pp 382-

Varga, R.S. Matrix iterative analysis, Prentice Hall, New

York, 1962.

Dr. Omar Ali Aleyan

31السنة العدد 454

Young, D. Iterative solution of large linear systems,

Academic Press, New York, 1977.

Richard L. Burden and J. Donglas Faires. Numerical

analysis, third edition prindle, Weber & Schmidt,

Boston, 1985.

المجلة ومعايير النشر مجة الجاعة الأسمسية دوزي ة عي ة داعة محه ة، تصدز ع الجاعة الأسمسي ة يعو الإضلاي ة

بيبيا، وتع عى ػس بحوخ أعطاء يئة ايت دزيظ الجاعي ، وػاطات ايعية ، وتأخر بأضباب

ا الجاعات الأخسى، ع طسيل د دطوز اي وض بايبخح ايعي ، وتطعى إلى زبط الجاعة بغير

ايج كافة والمعسفة، وخل كاط ايتكاء يتعاو فيا بيا، وبين عائسا المجلات، وبخاصة الجاعية

ا.

ايبخوخ المػوزة فيا تعبر ع آزاء أصخابا فكط، و وحد ايري يتخو المطؤويية ايكاوية

وآزائ، وصخة طبتا إلى صادزا، وييطت المجة طؤوية ع غيء ذيو، والأدبية ع أفهاز

ولا يص ػسا أ تهو عبرة ع ودة عسا.

ايبخوخ المكدة يذة حكا، لا تسد إلى أصخابا ػست أو لم تػس، ولا يجوش ػسا، أو

نتابي، إدازة ايتخسيس، ويجوش إعلا الاقتباع ا، أو تكديما يػس إلا بعد الحصو عى إذ

ايباحح بتيذة تكوي بحج، إذا طب ذيو بعد سوز غسي عى الأق تطي ايبخح، نا تخطع

ايبخوخ ايصالحة يػس يطياضة المجة في تطيل تستيبا، وفي ش ػسا.

هسة أو أصية، تػه إضافة يػترط في ايبخوخ ايتي تػس ألا تهو ػوزة قب، وأ تهو بت

وعي ة في اختصاصا، وتتوافس فيا الأصاية وايعل وصخ ة الأضوب، تصة بايكي الإطاي ة وبمعايير

ايبخح ايعي ، ولا ضي ا الابتعاد ع ايتذسيح والإضفاف في ايكو، وايت عسيض بالآخسي، زوعيت فيا

سادع، وتستيب المعوات بطل واحد في ايبخح، وتسقي ايبية المذي ة، واضتخدا المصادز والم

لات يبخح خسائط، أو دداو الهواؼ بأزقا طتك ة ع المصادز والمسادع، وإذا نات اى ه

فيبغي أ تهو في صوزتا الأصي ة، وإذا نا ايبخح ترجما يصخب بأص المترد ع.

ت كد ايبخوخ لإدازة المج ة طختين طبوعة عى وزم، ومحفوظة في قسص حاضوب، ويمه

إزضالها عى عوا الجاعة الإيهتروي ، ويفط إزضا المعوات المتعكة بايط يرة ايعي ة يباحح، في

وزقة طتك ة ع ايبخح.

تاة، تسفع الحسج ع ايباحح والمكو، ي عسض ايبخح عى كو تخصص، وكو يغوي في ضسي ة

وتعو يئة ايتخسيس نجيرا عى توصيات المكوين فيا يتعل بػس ايبخح عد.

د يػس، نا يطتخل المكو والمترد كابلا عى ايتكوي يطتخل ايباحح كابلا ايي ا ع بحج المعت

أو ايترجمة.

يبخوخ، بماقػتا وإثسائا وايسد ايعي عىى ىا وزد فيىا، وتفىتح صىدزا تسحب يئة ايتخسيس، بعد ػس ا

كد ايباء، وبخاصة المتخصصين، وتعد بأ ا يسد إييا يهو وضع ايعاية وايتكديس . لاضتكبا اي