11Dafik's Thesis M.Sc. OK

A STUDY OF MULTIGRID
METHODS WITH APPLICATION TO
CONVECTION-DIFFUSION
PROBLEMS
A dissertation submitted to the University of Manchester
Institute of Science and Technology for the degree of Master
of Science

April 1998
By
DAFIK
Department of Mathematics

For my wife, LIS, and son, OZI

ii

Contents
List of Tables

vi

List of Figures

viii

Abstract

ix

Declaration

x

Acknowledgements

xi

1 Introduction

1

1.1 The Convection-Diusion Problem . . . . . . . . . . . . . . . . .

3

1.2 The Galerkin Weak Formulation . . . . . . . . . . . . . . . . . . .

4

1.3 The Finite Element Formulation . . . . . . . . . . . . . . . . . . .

6

2 Stationary Iterative Methods

10

2.1 The Jacobi and Gauss-Seidel Methods . . . . . . . . . . . . . . . 10
2.2 The Damped Jacobi and Symmetric Gauss-Seidel Methods . . . . 16

3 Multigrid Methods

20

3.1 The Multigrid Idea . . . . . . . . . . . . . . . . . . . . . . . . . . 20
iii

3.1.1 Prolongation or Interpolation . . . . . . . . . . . . . . . . 22
3.1.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 The Multigrid Algorithm . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.1 The Basic Two-Grid Algorithm . . . . . . . . . . . . . . . 24
3.2.2 Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Complexity and Convergence Analysis . . . . . . . . . . . . . . . 27
3.3.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.2 Convergence analysis . . . . . . . . . . . . . . . . . . . . . 28
3.4 The Eectiveness of the Relaxation Methods . . . . . . . . . . . . 30

4 Numerical Results

35

4.1 Numerical Solution of the Diusion Problem . . . . . . . . . . . . 36
4.1.1 Convergence Rate of Simple Relaxation Methods . . . . . 37
4.1.2 Convergence Rate of a Representative Multigrid Met-hod . 39
4.2 Numerical Solution of the Convection-Diusion Problem . . . . . 42
4.2.1 Convergence Rate of Relaxation Methods . . . . . . . . . . 46
4.2.2 Convergence Rate of Multigrid Methods . . . . . . . . . . 48
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A The V-Cycle Multigrid Program

iv

53

List of Tables
4.1 General convergence for Gauss-Seidel method. . . . . . . . . . . . 37
4.2 General convergence for damped Jacobi method when ! = 4=5. . 38
4.3 General convergence for symmetric Gauss-Seidel method. . . . . . 38
4.4 General convergence for V{Cycle method with Gauss-Seidel as

smoother. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 General convergence for V{Cycle method with damped Jacobi
as smoother when ! = 4=5. . . . . . . . . . . . . . . . . . . . . . 40
4.6 General convergence for V{Cycle method with symmetric GaussSeidel as smoother. . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.7 Convergence rate of symmetric Gauss-Seidel method applied to the
problem with viscosity parameter 1/10. . . . . . . . . . . . . . . . 46
4.8 Convergence rate of the symmetric Gauss-Seidel method applied
to the problem with viscosity parameter 1/100. . . . . . . . . . . 47
4.9 Convergence rate of symmetric Gauss-Seidel method applied to the
problem with viscosity parameter 1/1000. . . . . . . . . . . . . . . 47
4.10 Convergence rate of V-Cycle method applied to the convectiondiusion problem with viscosity parameter 1/10. . . . . . . . . . . 49

v

4.11 Convergence rate of V-Cycle method applied to the convectiondiusion problem with viscosity parameter 1/100. . . . . . . . . . 49
4.12 Convergence rate of V-Cycle method applied to the convectiondiusion problem with viscosity parameter 1/1000. . . . . . . . . 50
4.13 Convergence rate of V-Cycle method applied to the convectiondiusion problem with \optimal" restriction coecient. . . . . . . 51

vi

List of Figures
1.1 Rectangular Element. . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.1 Grids schedule for a V-Cycle with ve levels. . . . . . . . . . . . 26
4.1 The diusion problem solution for a (64 64) uniform grid with
tolerance  =1e-6. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 The convergence history of the relaxation methods when the tolerance  =1e-6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 The convergence history of V-Cycle method with Gauss-Seidel,
damped Jacobi, and symmetric Gauss-Seidel as smoothers when
the tolerance  =1e-6. . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 The solution for the viscosity parameter 1/10. . . . . . . . . . . . 43
4.5 The solution for the viscosity parameter 1/100. . . . . . . . . . . 44
4.6 The solution for the viscosity parameter 1/1000. . . . . . . . . . . 44
4.7 The convergence history of symmetric Gauss-Seidel applied in the
convection-diusion problem without upwinding. . . . . . . . . . . 48
4.8 The convergence history of symmetric Gauss-Seidel ( =1e-6) applied in the convection-diusion problem. . . . . . . . . . . . . . . 50

vii

4.9 The convergence history of V{Cycle method with symmetric
Gauss-Seidel as smoother ( =1e-6) applied in the convectiondiusion problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

viii

Abstract
The work accomplished in this dissertation is concerned with the numerical solution of linear elliptic partial dierential equations in two dimensions, in particular
modelling diusion and convection-diusion. The nite element method applied
to this type of problem gives rise to a linear system of equations of the form

Ax = b. It is well known that direct and classical iterative (or relaxation) methods can be used to solve such systems of equations, but the eciency deteriorates
in the limit of a highly rened grid. The multigrid methodologies discussed in this
dissertation evolved from attempts to correct the limitations of the conventional
solution methods. The aim of the project is to investigate the performance of
multigrid methods for diusion problems, and to explore the potential of multigrid in cases where convection dominates.

ix

Declaration

No portion of the work referred to in this dissertation has been submitted in
support of an application for another degree or qualication of this or any other
University or other institution of learning.

x

Acknowledgements
Throughout the preparation of this dissertation, I have beneted from the kind
assistance of my supervisor Dr. D.J. Silvester. I would like to take this opportunity to express my gratitude for all the help and guidance I have been given.
Also I would like to thank Dr. David Kay for his help in mastering the Finite
Element Method.
I would also like to thank all my family, especially my parents, my wife and
my sponsor (DUE Project of Jember University) who have always encouraged my
academic endeavours and have been a constant source of inspiration.
My thanks are also due to the academic and secretarial sta here at the
University of Manchester Institute of Science and Technology and also at the
Victoria University of Manchester for all their help.

xi

Contents

xii

List of Tables

xiii

List of Figures

xiv

Chapter 1
Introduction
In this project we consider the numerical solution of linear elliptic partial dierential equations in two dimensions, in particular modelling di usion and convectiondi usion. We will use the nite element discretisation method, see Johnson 3],

for details, which generates a system of linear equations of the form Ax = b, where

A is a sparse, positive denite matrix (and is symmetric in the diusion case). It
is well known that such systems of equations can be solved eciently using direct

and iterative (or relaxation) methods. In the diusion case the preconditioned
conjugate gradient method is another possibility.
Direct methods, of which Gaussian elimination is the prototype, determine a
solution exactly (up to machine precision and assuming a perfectly conditioned
matrix) in a nite number of arithmetic steps. They are often based on the
fast Fourier transform or the method of cyclic reduction. When applied to PDE
problems discretised on an N N grid, these methods require O(N 2 logN ) arithmetic operations. Therefore, since they approach the minimum operation count
of O(N 2 ) operations, these methods are nearly optimal. However, they are also
1

CHAPTER 1. INTRODUCTION

2

rather specialised and can be applied primarily to a system which arises from
separable self-adjoint boundary value problems.
Relaxation methods, as represented by the Jacobi and Gauss-Seidel iterations,
begin with an initial guess at solution. They then proceed to improve the current approximation by a succession of simple updating steps or iterations. The
sequence of approximations that is generated (ideally) converges to the exact solution of the linear system. Classical relaxation methods are easy to implement
and may be successfully applied to more general linear systems than the direct

methods.
However, these relaxation schemes also suer from some disabling limitations.
Multigrid methods evolved from attempts to correct these limitations. These
attempts have been largely successful used in a multigrid setting, relaxation
methods are competitive with the fast direct methods when applied to the model
problems.
The basic multigrid methods also have immediate extensions to boundary
value problems with more general boundary conditions, operators and geometries.
It is safe to state that these methods can be applied to any self-adjoint (symmetric
positive denite) problem with no signicant modication.
There is actually no simple answer to the question of whether direct or iterative methods are ultimately superior. Since it depends upon the structure
of the problem and type of computer being used. We should at least consider
the grid size, the number of iterations taken to converge, the approximate work
units, and also the computer CPU times, to assess whether one method is more
eective than the others. These factors are also to be an indicator in determining

CHAPTER 1. INTRODUCTION

3

the eciency of those methods.
The basic multigrid methods are certainly not conned to nite dierence
formulations. In fact, nite element discretisation are often more natural, particularly for analysis of these methods. Herein, we will use the nite element
method to discretise the model diusion and convection-diusion problems, and
then solve the resulting discrete system using multigrid methods. The eectiveness and eciency will be compared with classical relaxation methods.

1.1 The Convection-Diusion Problem
The general test problem that we consider is the convection-diusion problem in
a region 
IR2

;r2u + w  ru = f in 

(1.1)

where  is viscosity parameter, ;r2u represents diusion (dened by ;( @@x2u2 +
@2u
@y2

@u
)) and w ru represents convection (and is given by !x @x
+ !y @u
@y ), see Morton

1], for details. The conditions on the boundary of  are dened by
u = 0 on @ d 6= 
@u
= 0 on @ n 
@n
@u
is the normal derivative, and  is the region with the boundary @ d 
where @n

@ n = @  and @ d \ @ n = . The boundary conditions u = 0 are known as the

Dirichlet boundary conditions, and

@u
@n

= 0 as Neumann conditions. We wish to

nd an approximation to the solution u of this problem. To do this we will use
the Galerkin method with a specic choice of basis functions.

CHAPTER 1. INTRODUCTION

4

1.2 The Galerkin Weak Formulation
The weak formulation of equation (1.1) is to nd u V such that
2

a1(u v) + a2(u v) = (f v)
where

Z
Z
Z

a1(u v) =



a2(u v) =



(f v) =



 u
r

(w

8

r

r

v V
2

v d

u)v d

fv d

and

V =   H 1()  = 0 on @ d 
f

j

2

g

where the Sobolev space H 1() is the space of functions with square integrable
rst derivatives.
To nd an approximation to u, we choose an m dimensional subspace Vm of

V given by
Vm = v v =
f j

Xm ii i
i=1

2

IR

 i V
2

g

i = 1 2 : : :  m:

The Galerkin approximation um from the subspace Vm satises

a1(um v) + a2(um v) = (f v)

8

v Vm :

(1.2)

2

Since um Vm we may write um in the form
2

m
X
u m =  i i 
i=1

and thus we need to nd the i

2

IR

i

2



(1.3)

IR

for i = 1 2 : : :  m. Since j
f

m
j =1

g

are the

basis functions for Vm , we need only to test over j  j = 1 2 : : :  m in equation

CHAPTER 1. INTRODUCTION

5

(1.2). Therefore, the equation (1.2) is equivalent to nding um 2 Vm such that

a1(um j ) + a2(um j ) = (f j ) 8j = 1 2 : : :  m:

(1.4)

Substituting (1.3) into (1.4) gives
m
X

m
X

i=1

i=1

a1( ii j ) + a2( ii j ) = (f j )
and recalling the weak formulation we have
Z




m
X
i=1

!

iri  rj d +

Z


w

m
X
i=1

!

iri j d =

Z


fj d:

We denote

Aij =
(1)

A(2)
ij =
bj =

Z
Z



Z




ri  rj d
(w  ri)j d

fj d

where A(1) A(2) are matrices which arise from the diusion and convection terms
respectively, and b is called the force vector and
i @j @i @j
ri  rj =  @
@x @x + @y @y
@i  :
i
(w  ri)j = wx @

j + wy
@x
@y j

The resulting linear system is of the form

A = b
where A is an m m matrix dened by A := A(1) + A(2).

!

CHAPTER 1. INTRODUCTION

6

1.3 The Finite Element Formulation
The basic idea of the nite element method is to use a piecewise polynomial
approximation. The solution process, referring to Johnson 3], can be divided
into steps as follows
1. Split the region of interest into rectangular (or triangular) elements K.
2. Dene a local polynomial (for approximating the solution) within each element.
3. Dene a set of nodes in each element.
4. Dene the local polynomials in terms of the nodal values.
5. Satisfy the essential boundary conditions.
6. Solve the resulting discrete set of algebraic equations.
Since a bilinear function, in x and y is of the form

(x y) = p + qx + ry + sxy

(1.5)

we see that there are four unknown parameters which can be uniquely determined
by values at four nodes (x1 y1) (x2 y2) (x3 y3) (x4 y4) (no more than two in a
straight line). The bilinear function can then be written in the general form

(x y) = 1(x y) 1 + 2(x y) 2 + 3(x y) 3 + 4(x y) 4
where the  's are the nodal basis function satisfying
i

8
>< 1 if i = j
 (x  y ) = >
: 0 if i = j i j = 1 : : :  m
j

i

i

6

CHAPTER 1. INTRODUCTION
that is (

i x y

of

7

) = ^ + ^ + ^ + ^ . These functions are obtained by the solution
p

qx

2
66
66
66
66
66
64

ry

sxy

1

x1

y1

x 1 y1

1

x2

y2

x 2 y2

1

x3

y3

x 3 y3

32
6
7
6
7
6
7
6
7
6
7
6
7
6
7
6
7
6
7
6
7
7
4
56

^

p

^

q

^

r

3 2
6
7
6
7
6
7
6
7
6
7
6
7
6
=
7
6
7
6
7
6
7
7
4
5 6

ei

3
7
7
7
7
7
7
7
7
7
7
7
5

^
1 4 4 44
where = 0 except ( ) = 1. This system can be solved provided that not more
ei

x

y

x y

s

ei i

than two of the nodes lie in a straight line.
Now, given the simplest rectangular element with just four nodes, one at each
corner, we choose the local coordinate (


 

) as shown in gure 1.1.

ψ
4

ψ
3

(-1,1)

(1,1)
η=

l

(x m,y m)

l

(y - y m )

ξ=

(-1,-1)
ψ1

2

2

k

(x - x m )

(1,-1)

k

ψ2

Figure 1.1: Rectangular Element.
Since there are four nodes with one degree of freedom at each node the variation throughout the element is just the bilinear form given in (1.6) below

CHAPTER 1. INTRODUCTION

8

(x y) = 

= 41 (1 ;
)(1 ; ) (1 +
)(1 ; ) (1 +
)(1 + )
2 3
66 1 77
66 77
6 2 7
(1 ;
)(1 + )] 666 777 :
66 3 77
64 75

(1.6)

4

We have

@ = 2 @ and @ = 2 @ :
@x k @

@y l @
Now we can obtain the stiness matrices A(1) A(2) and force vector b of the
convection-diusion system, say for the special case of constant f (x y) = c. The
calculation can be shown by taking an example

A =
(1)
11

Z1Z1
;1

!

4 @1 @1 + 4 @1 @1 k @
l @:

l2 @ @ 2 2
;1 k 2 @
@


Substituting 1 = 14 (1 ;
)(1 ; ) and doing the integrating process we get
1
A(1)
11 = (k=l + l=k ):
3
By repeating the above process for all entries and noting symmetry for the
diusion matrix we obtain

3
2
66 2(k=l + l=k ) k=l ; 2l=k ;k=l ; l=k l=k ; 2k=l 77
77
66
77
6
2(
k=l
+
l=k
)
l=k
;
2
k=l
;
k=l
;
l=k
A(1) = 61  666
77
66 symmetric
2(k=l + l=k) k=l ; 2l=k 77
75
64

2(k=l + l=k)

CHAPTER 1. INTRODUCTION

9

whilst entries for the convection matrix are obtained by integrating

A =
(2)
ij

!

Z1Z1

2 @  k @
l @
w k2 @

+
w
@

l @ 2 2
i

;1 ;1

x

i

j

y

leading to the matrix

2
66 ;lw ; kw
66
66 ;lw ; 21 kw
1
(2)
A = 6 66
66 ; 21 lw ; 21 kw
64
x

lw ; kw
lw ; kw
1
2 lw ; kw
1
1
2 lw ; 2 kw

y

x

x

y

x

; 21 lw ; kw
x

1
2

x

y

y

3
lw + kw ; lw + kw 77
77
1
1
1
77
lw
+
kw
;
lw
+
kw
2
2
2
77 :
1
lw + kw ;lw + 2 kw 777
5

1
2

y

y

x

x

x

y

x

j

x

y

1
2

y

y

y

lw + 12 kw
x

y

1
2

x

y

x

x

;lw + kw
x

y

y

y

The overall element stiness matrix is then obtained by the sum of A(1) A(2) i.e.

A = A(1) + A(2).
The force vector has entries obtained by

b =
j

leading to the vector

Z1Z1
;1

k @
l @
c
2 2
;1
j

2 3
66 1 77
66 77
66 1 77
b = klc
4 666 1 777 :
66 77
4 5

1
Assembling the element contributions yields the overall system of equations.

Chapter 2
Stationary Iterative Methods
2.1

The Jacobi and Gauss-Seidel Methods

The simplest iterative scheme is the Jacobi iteration. It is dened for matrices
that have nonzero diagonal elements, in particular diagonally dominant matrices.
This method can be motivated by rewriting the n n system Ax = b as follows:

x1 = (b1 ; a12x2 ; a13x3 ; : : : ; a1 x )=a11
n

n

x2 = (b2 ; a21x1 ; a23x3 ; : : : ; a2 x )=a22
n

n

...

x = (b ; a 1x1 ; a 2x2 ; : : : ; a
n

n

n

n

nn

;1 xn;1 )=ann :

In the Jacobi method, we choose an initial vector x(0) and then solve, iteratively,
to nd a new approximation x( +1) using the algorithm
i

x(1 +1) = (b1 ; a12x(2 ) ; a13x(3 ) ; : : : ; a1 x( ))=a11
i

i

i

n

i
n

x(2 +1) = (b2 ; a21x(1 ) ; a23x3( ) ; : : : ; a2 x( ))=a22
i

i

i

n

...

10

i
n

CHAPTER 2. STATIONARY ITERATIVE METHODS
x(ni+1)

= (b ; a
n

n1

( i)

x1

; a 2x(2 ) ; : : : ; a
i

n

nn

11

;1 xn;1 )=ann :
( i)

We notice here that Jacobi iteration is not updating the most recently available
information when computing x( +1), where k is any entry of x( +1) after the rst
i

i

k

one. For an example, if we look at the x(2 +1) calculation we observe that x(1 ) is
i

i

used even though we know x(1 +1). If we use this most recent estimate we form
i

the Gauss-Seidel iteration
(i+1)

= (b1 ; a12x(2 ) ; a13x(3 ) ; a14x(4 ) ; : : : ; a1

(i+1)

= (b2 ; a21x(1 +1) ; a23x(3 ) ; a24x(4 ) ; : : : ; a2

(i+1)

= (b3 ; a31x(1 +1) ; a32x(2 +1) ; a34x(4 ) ; : : : ; a3
...

x1

x2
x3

x(ni+1)

i

i

i

n

i

i

i

x(ni))=a11

i

i

n

x(ni))=a22

i

n

= (b ; a
n

n1

(i+1)

x1

; a 2x(2 +1) ; : : : ; a
i

n

nn

x(ni))=a33

(i+1)
;1 xn;1 )=ann :

We use x(1 +1) to calculate x(2 +1) x(2 +1) to calculate x(3 +1) and hence use x( ;+1)
1 to
i

i

i

i

i
n

calculate x( +1). The method which updates the most recently available calculai
n

tion tends to converge faster than the method which does not (see later).
We now let L D, and U be matrices dened as
2
0
6
6
6
6
6
a21
6
6
6
L = ; 6 a31
6
6
6
...
6
6
6
4

0

:::

0 ...
... ...
...

...

0
...
...

0

0

:::

an1 : : : : : : ann;1

0

3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5

CHAPTER 2. STATIONARY ITERATIVE METHODS
2
6
6
6
6
6
6
6
D = 666
6
6
6
6
6
4

U

2
6
6
6
6
6
6
6
= ; 666
6
6
6
6
6
4

a11 0 : : : : : :
0 a22 . . .
... . . . a . . .
33
...

0
...
...

... ... 0

0 : : : : : : 0 ann
0 a12 : : : : : :
0 0 ...
... . . . . . . . . .
...
... 0

12

3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5

3
a1n 77
... 777
7
... 777 
7
7
7
an;1n 777
5

0 ::: ::: 0 0
so that A = D ; L ; U . The Jacobi method is of the form

x(i+1) = Hx(i) + cJ 

(2.1)

with H = D;1 (L + U ) and cJ = D;1 b. The Gauss-Seidel step has the dierent
form of

x(i+1) = Gx(i) + cG 

(2.2)

with G = (D ; L);1 U and cG = (D ; L);1b.
Both procedures are typical members of a large family of iterations that have
the form

Qx(i+1) = V x(i) + b:

(2.3)

Whether or not (2.3) converges to x = A;1b depends upon the eigenvalues of

Q;1V . In particular, if the spectral radius of an n-by-n matrix Z is dened by
(Z ) = maxfj
j
2
(Z )g

CHAPTER 2. STATIONARY ITERATIVE METHODS

13

then it is the size of (Q;1V ) that determines the convergence of (2.3). We now
introduce the denition of a non-negative matrix.

Denition 2.1.1 Let A = (a ) and B = (b ) be two n n matrices. Then,
A
B (> B ) if a
b (> b ) for all 1  i j  n. If O is the null matrix
and A
O(> O), we say that A is a non-negative (positive) matrix. Finally,
if B = (b ) is an arbitrary real or complex n n matrix, then jB j denotes1 the
matrix with entries jb j.
ij

ij

ij

ij

ij

ij

ij

A mutually exclusive relation between the Jacobi matrix H and the Gauss-Seidel
matrix G is given in the following theorem developed by Varga 5, pp.70].

Theorem 2.1.1 Let the Jacobi matrix H = E + F (where E = D;1 L and F =
D;1 U ) be a non-negative n n matrix with zero diagonal entries, and G =
(I ; E );1F . Then, one and only one of the following mutually exclusive relations
is valid:
1. (H ) = (G) = 0:
2. 0 < (G) < (H ) < 1:
3. 1 = (H ) = (G):
4. 1 < (H ) < (G):
Thus, the Jacobi and the Gauss-Seidel iterations are either both convergent, or
both divergent.
1

This is not to be confused with the determinant of a square matrix B

CHAPTER 2. STATIONARY ITERATIVE METHODS

14

Denition 2.1.2 An n n real or complex matrix A = (a ) is diagonally domij

inant if

X
ja j
ja j
n

ii

=1
=i

j
j

for all 1  i  n

ij

(2.4)

6

and is strictly diagonally dominant if strict inequality in (2.4) is valid for all

1  i  n. Similarly, A is irreducibly diagonally dominant if A is irreducible
(see Varga 5, pp.18-19], for denition of the reducible matrix) and diagonally
dominant, with strict inequality in (2.4) for at least one i.

Theorem 2.1.2 Suppose b 2 IR and A = Q;V 2 IR
is nonsingular. If Q is
n

n

n

non singular and the spectral radius of Q;1 V satises the inequality (Q;1V ) < 1,
then the iterates x( ) dened by Qx( +1) = V x( ) + b converge to x = A;1b for any
i

i

i

starting vector x(0).

Proof. Let e( ) = x( ) ; x denote the error in the ith iterate. Since Qx( +1) =
V x( ) + b it follows that Q(x( +1) ; x) = V (x( ) ; x), and thus, the error in x( +1)
is given by e( +1) = Q;1V e( ) = (Q;1V )( )e(0). We know that (Q;1V )( ) ! 0
i

i

i

i

i

i

i

i

i

i

i

i (Q;1V ) < 1, see Golub 4, pp.508]. So x( ) dened by Qx( +1) = V x( ) + b
i

i

i

converge to x = A;1b for any starting vector x(0).
Another fundamental theorem is concerned with the convergence of the strictly
diagonally dominant matrix.

Theorem 2.1.3 Let A = (a ) be a strictly or irreducibly diagonally dominant
ij

n n real or complex matrix. Then, both Jacobi and Gauss-Seidel methods are
convergent to x = A;1 b for any starting vector x(0).

Proof. If A is a n n strictly diagonally dominant real or complex matrix, then it
is nonsingular, see Varga 5, pp.23], and the diagonal entries of A are necessarily

CHAPTER 2. STATIONARY ITERATIVE METHODS

15

nonzero. From (2.1) the Jacobi matrix H derived from the matrix A is such that

8
>< 0
hij = >
: ;aa

if i = j
ij

ii

if i 6= j:

By Denition 2.1.2 it follows that
n
X
j =1

jhij j < 1 for all 1  i  n

if A strictly diagonally dominant, or
n
X
j =1

jhij j  1 for all 1  i  n

with strict inequality for at least one i if A is irreducibly diagonally dominant.
Recalling Denition 2.1.1 jH j is a non-negative matrix whose entries are jhij j, and
these row sums tell us that (jH j) < 1. Applying Theorem 2:8 in 5], we have
that for any n n matrices A and B with O  jB j  A, then (B )  (A). We
can conclude here that (H )  (jH j) < 1, so that the Jacobi iterative method
is necessarily convergent. For the Gauss-Seidel matrix G we have

G = (D ; L);1U or G = (I ; E );1F
where E and F are respectively strictly lower and upper triangular matrices.

G = (I ; E );1F

jGj = j(I + E + E 2 + E 3 + : : : + E n;1 )F j
 (I + jE j + jE j2 + jE j3 + : : : + jE jn;1 )jF j
= (I ; jE j);1jF j:
So that (G)  f(I ; jE j);1jF jg. But as (jH j) < 1, we have from Theorem
2.1.1 that f(I ; jE j);1jF jg  (jH j) < 1. This implies

(G) < 1

CHAPTER 2. STATIONARY ITERATIVE METHODS

16

thus we conclude that the Gauss-Seidel iterative method is also convergent, which
completes the proof.

2.2 The Damped Jacobi and Symmetric GaussSeidel Methods
There is a simple but important modication to the Jacobi iteration. This generates an entire family of iterations called a weighted or damped Jacobi method.
The formula is given by

x(i+1) = (1 ; !) + !H ]x(i) + !cJ 

(2.5)

where ! 2 IR is a weighting factor which may be chosen. Notice that with ! = 1
we have the original Jacobi iteration.
We should note in passing that the weighted Jacobi iteration can also be
written in the form

x(i+1) = x(i) + !D;1 r(i):
This says that the new approximation is obtained from the current one by adding
an appropriate weighting of the residual r(i) = b ; Ax(i). In further prediction,
this method tends, with a well chosen !, to converge faster than Jacobi method.
In general, we know that the error e(i) = x(i) ; x satises Ae(i) = r(i), so we have

x(i) ; x = A;1r(i). From this expression, an iteration may be performed by taking
x(i+1) = x(i) + Br(i), where B is some approximation to A;1. If B can be chosen
\close" to A;1, then the iteration should be eective.
The weighted Jacobi method waits until all components of the new approximation have been computed before using them. This requires 2N storage locations

CHAPTER 2. STATIONARY ITERATIVE METHODS

17

for the approximation vector. It means that the new approximation can not be
used as soon as it is available. On the contrary, the Gauss-Seidel method incorporates a simple change: components of the new approximation are used as soon as
they are computed, so that components of the approximation vector x are overwritten as soon as they are updated. Moreover, for the weighted Jacobi method,
the order of updating the components (ascending or descending) is immaterial,
since they are never over-written. However, for the Gauss-Seidel method, the
order is signicant.
The Gauss-Seidel method normally generates a nonsymmetric iteration matrix. This can be demonstrated by taking a 2 2 system as an example,
2
66 a11 a12
4

a21 a22

3 2 3
32
x1 7 6 b1 7
6
7
7
6
7
5:
4 7
5=6
54
b2

x2

If we perform one iteration of Gauss-Seidel,
(1)

=

b1

; a12x(0)
2

(1)

=

b2

; a21x(1)
1 

a11x1

a22x2

(1)
and if x(0) is zero, then x(1)
1 = b1 =a11 and x2 = b2=a22 ; a21b1 =a11a22 so

2
6
x(1) = 6
4

This is equivalent to x(1) =

Kb.

1

a11

;a aa

21
11 22

0
1

a22

32 3
6 b1 7
7
7
4 7
56
5:
b2

The matrix K here is not symmetric except

for the trivial case when a21 is zero. However, we also have a symmetric version
of the Gauss-Seidel method. If we perform one iteration of Gauss-Seidel again,
initialising x(0) to zero, this time with a 3 3 system.
a11x1

=

b1

(0)
; a12x(0)
2 ; a13x3

CHAPTER 2. STATIONARY ITERATIVE METHODS
a22x2

=

b2

; a21x1 ; a23x(0)
3

a33x3

=

b3

; a31x1 ; a32x2:

18

Therefore
x1

=

x2

= (b2 ; a21b1=a11)=a22

x3

= b3 ; a31b1=a11 ; a32(b2 ; a21b1=a11)=a22]=a33:

b1 =a11

(2.6)

If we then compute x(1) using x as an estimate, via
(1)

=

b3

; a31x1 ; a32x2

(1)

=

b2

; a21x1 ; a23x(1)
3

(1)

=

b1

(1)
; a12x(1)
2 ; a13x3 

a33x3

a22x2

a11x1

this gives our new solution value of x(1). If we write x(1) = Kb, we can show that
K

is symmetric and positive denite. Rearranging, we obtain
(1)

a11x1

(1)
+ a12x(1)
=
2 + a13x3

b1

+ a23x(1)
=
3

b2

; a21x1

=

b3

; a31x1 ; a32x2:

(1)

a22x2

(1)

a33x3

This shows us that what we have performed here is a lower triangular sweep
followed by an upper triangular sweep. This is the same as writing
(D ; U )x(1) = b + Lx:
From the set of equation (2.6), we have
b1

=

a11x1

b2

=

a21x1 + a22x2

b3

=

a31x1 + a32x2 + a33x3

(2.7)

CHAPTER 2. STATIONARY ITERATIVE METHODS

19

which can be written as (D ; L)x = b. Hence x = (D ; L);1b, and substituting
for x in equation (2.7) gives (D ; U )x(1) = b + L(D ; L);1b.
Therefore

x(1) = (D ; U );1(I + L(D ; L);1)b
= (D ; U );1((D ; L) + L)(D ; L);1b
= (D ; U );1D(D ; L);1b:
Dene K = (D ; U );1D(D ; L);1. To show the symmetry,

K T = (D ; L)T ];1DT (D ; U )T ];1
but (D ; L)T = (D ; U ) (A is a symmetric matrix in the diusion case) which
implies that K T = (D ; U );1 D(D ; L);1. Hence K is symmetric. For positive
deniteness

xT Kx = xT (D ; U );1D(D ; L);1x
and if we let y = (D ; L);1x and y 6= 0 provided x 6= 0 then

yT = xT (D ; L)T ];1
= xT (D ; U );1 :
Therefore

xT Kx = yT Dy > 0:
(Since A is positive denite then D is a diagonal matrix with positive denite
diagonal elements). Thus K is positive denite. For more details about symmetric
Gauss-Seidel, the reader is referred to Golub 4].

Chapter 3
Multigrid Methods
3.1

The Multigrid Idea

We now introduce the standard multigrid idea for the solution of the algebraic
equations arising from nite element discretisation of dierential equations. We
thus look to multigrid algorithms to provide an ecient iteration for solving a
large, sparse system of equations. The eciency of multigrid algorithms arise from
the interaction between the smoothing properties of the relaxation method, such
as Gauss-Seidel iteration, and the idea of coarse grid correction, see McCornick
12].
Using standard local Fourier analysis it can be seen that relaxation methods,
without convergence acceleration parameters, are very ecient at reducing the
amplitude of high-frequency components of the error, and thus of the algebraic
residual, whilst their ultimately slow convergence is due to the low-frequency
components, typically corresponding to the largest eigenvalues of the iteration
operator. Relaxation methods can thus be viewed as ecient smoothers.
20

CHAPTER 3. MULTIGRID METHODS

21

The principle of coarse grid correction can be seen by considering linear nite
element equations. Denoting the grid by h , we let G(h ) be the space of all
grid functions dened on h. Then, assuming that the boundary conditions have
been eliminated, the linear equations can be written in the form

Ahxh = bh
where xh bh 2 G(h ) Ah : G(h ) ! G(h ) and we assume that (Ah);1 exists.
Let vh be the current approximation to xh and dene the algebraic defect (or
residual) as

rh = bh ; Ahvh:
Then the exact solution xh is given by

xh = vh + sh
where sh is the solution of the defect equation

Ahsh = rh:

(3.1)

If the high-frequency components of the defect are relatively negligible to the low
frequency components, we can represent equation (3.1) on a coarser grid, H (in
practice, we dene a sequence consisting of 2h 4h : : :) of the form:

AH s^H = rH

(3.2)

where dim(G(H ))
dim(G(h)) AH : G(H ) ! G(H ) and we assume that
(AH );1 exists.
If we solve the equation (3.2) we can interpolate s^H to the ner grid giving
an approximation s^h to sh and then take

v^h = vh + s^h

CHAPTER 3. MULTIGRID METHODS

22

as the new approximation to xh. The assumption here is that the approximation
to the solution on the ne grid should be smooth enough to allow the equation
to be represented on a coarse grid. This criterion is satised by using a suitable
relaxation method before transferring the defect equation to the coarser grid.
The multigrid method extends this idea to the solution of the coarse grid
equation (3.2) leading to a series of coarser and coarser grids. Thus the complete
multigrid method combines a relaxation method on each grid with correction on
coarser grids. Neither process alone gives a satisfactory method it is only when
a suitable combination of the two methods is used that good convergence and
eciency properties can be obtained.
The remaining issues are the computation of the residual on h and it's transfer to 2h, and the computation of the error estimate on 2h and it's transfer to
h . These questions suggest that we need some mechanism for transferring information between grids. The step in the coarse grid correction scheme that requires
transferring the error approximation e2h from the coarse grid 2h to the ne grid
h is generally called interpolation or prolongation. Secondly, the inter-grid transfer function which involves moving the vectors from a ne grid to a coarse grid
is generally called restriction operator.
3.1.1 Prolongation or Interpolation

The linear interpolation denoted by I2hh takes coarse grid vectors and ne grid
vectors according to the rule I2hhv2h = vh. For one-dimensional problems, they
may be dened as

CHAPTER 3. MULTIGRID METHODS

=

h

v2j

2h

vj

= 21 (

h

v2j +1

23

2h

vj

+

2h

where 0   2 ; 1

)

vj +1 

N

j

:

At even-numbered ne grid points, the values of the vector are transferred directly
from 2 to  . At odd-numbered ne grid points, the values of the
h

h

v

h

are the

average of the adjacent coarse grid values. As for two-dimensional problems, they
may be dened in a similar way. If we let
v

h

h

I2h v

2h

= , then the components of
v

h

are given by
=

h

v2i
2j

2h

vi
j

= 21 (
= 12 (
= 14 (

h

v2i+1
2j
h

v2i
2j +1
h

v2i+1
2j +1

2h

+

vi+1
j

2h

+

vi
j +1

2h

+

vi+1
j

vi
j

vi
j

vi
j

2h

)

2h

)

2h

+

+

2h

vi
j +1

0

)

2h

vi+1
j +1 

i j

 2 ;1
N

:

3.1.2 Restriction

The restriction operator is denoted by
tion

dened (in one dimension) by

2h

Ih v

2h

vj

2h

Ih

=

h

. The most obvious restriction is injec-

=

v

2h

, where

h

v2j :

In other words, the coarse grid vector simply takes its value directly from the
corresponding ne grid point. An alternate restriction operator is called
weighting

and is dened (in one dimension) by
2h

vj

= 41 (

h

v2j

;1 + 2v2

h
j

+

h

)

v2j +1 

2h

Ih v

h

=

v

2h

, where

1  2 ;1
j

N

full

:

(3.3)

CHAPTER 3. MULTIGRID METHODS

24

In this case, the values of the coarse grid vector are weighted averages of values
at neighbouring ne grid points.
The full weighting restriction operator is a linear operator from IR

N

;1 to IR N2 ;1 .

It has a rank of N=2 ; 1 and null space with dimension N=2. One reason for our
choice of full weighting as restriction operator is the important fact that

I2 = c(I 2 ) 
h

h

h

(3.4)

T

h

where c = 2. Using the denition (3.3) we can also formulate the restriction
operator in two dimensions. Letting I 2 v = v2 , we have that
h

h

h

h

v2 = 161 v2 ;1 2 ;1 + v2 ;1 2 +1 + v2 +1 2 ;1 + v2 +1 2 +1 + 2(v2 2 ;1
+v2 2 +1 + v2 ;1 2 + v2 +1 2 ) + 4v2 2 ]
1  i j  N2 ; 1:
h

h

h

i

i
j


j

h

h

i


j

h

i
j

h

i


j

h

i


j

h

i


j

i
j

h

i


j

i
j

3.2 The Multigrid Algorithm
3.2.1 The Basic Two-Grid Algorithm
A well-dened way to transfer vectors between ne and coarse grids can be represented in the following coarse grid correction scheme

v  CG(v  b ):
h

h

h

So that the two-grid algorithm can be written as
Relax s1 times on A x = b on  with initial guess v :
h

h

h

h

h

Compute r2 = I 2 (b ; A v ):
h

h

h

h

h

h

Solve A2 e2 = r2 on 2 :
h

h

h

h

Correct ne grid approximation : v  v + I2 e2 .
h

h

h
h

h

CHAPTER 3. MULTIGRID METHODS

25

Relax s2 times on Ahxh = bh on h with initial guess vh:
The procedure dened above is simply the original coarse grid correction idea,
which was proposed earlier, now rened by the use of the intergrid transfer operators. We relax on the ne grid until it ceases to be worthwhile. In practice s1 is
often 1,2, or 3. The residual of the current approximation is computed on h and
then transferred by a restriction operator to the coarse grid. As it stands, the
procedure calls for the exact solution of the residual equation on 2h , which may
not be possible. However, if the coarse grid error can at least be approximated,
it is then interpolated up to the ne grid, where it is used to correct the ne grid
approximation. This is followed by s2 additional ne grid relaxation sweeps.
We can further expand this algorithm to higher grids level, say n levels. This
idea is fundamental to the understanding of multigrid.
3.2.2 Multigrid

The multigrid algorithm which is formulated in Briggs 10] can be expressed as
Relax on Ahxh = bh s1 times with initial guess vh.
Compute b2h = Ih2hrh.
Relax A2hx2h = b2h s1 times with initial guess v2h = 0.
Compute b4h = I24hhr2h .
Relax on A4hx4h = b4h s1 times with initial guess v4h = 0.
Compute b8h = I48hhr4h.
...
Solve ALhxLh = bLh.
...

CHAPTER 3. MULTIGRID METHODS

26

Correct v4h  v4h + I84hhv8h.
Relax on A4hx4h = b4h s2 times with initial guess v4h.

Correct v2h  v2h + I42hhv4h.

Relax A2hx2h = b2h s2 times with initial guess v2h.

Correct vh  vh + I2hhv2h.

Relax on Ahxh = bh s2 times with initial guess vh.
Figure 3.1 illustrates the grid schedule of multigrid method.
o

o

o

o

o

o

o

o

o

h

2h

4h

8h

16h

Figure 3.1: Grids schedule for a V-Cycle with ve levels.
The algorithm telescopes down to the coarsest grid, which can be a single
interior grid point, and then works its way back to the nest grid. Figure 3.1
shows the schedule for the grids in the order in which they are visited. Because of
the pattern of this diagram, this algorithm is called the V-cycle. It is our rst
true multigrid method. For more details of the algorithm, including a compact
recursive denition, the reader is referred to Briggs 10].

CHAPTER 3. MULTIGRID METHODS

27

3.3 Complexity and Convergence Analysis
3.3.1 Complexity
We now analyse a complexity and convergence of multigrid method. These points
are important to address due to the practical issues of the eectiveness and eciency of the overall method. The complexity is considered rst.
Each grid in the multigrid scheme needs two arrays: one to hold the current
approximation on each grid and one to hold the right-hand side vectors on each
grid, see McCormick 12]. Since boundary values must also be stored, the coarsest
grid involves three grid points in one dimension (one interior and two boundary
points). In general, the th coarsest grid involves 2k + 1 point in one dimension.
k

Now considering a ;dimensional grid with
d

grid h requires 2

N

d

N

d

points where

N

= 2n , the nest

storage locations 2h has 2;d times as many 4h has 2;2d

times as many etc. (by assumption two arrays must be stored on each level).
Adding these and using the sum of the geometric series as an upper bound give
us
Storage = 2

d

f1 + 2;d + 2;2d + : : : + 2;nd g <

2

N

d

1 ; 2;d
In particular, for a one-dimensional problem, the storage requirement is less than
N

:

twice that of the ne grid problem alone. For two or more dimensions, the
requirement drops to less than 4/3 of the ne grid problem alone. Thus, the
storage costs of the multigrid algorithm decrease relatively as the dimension of
the problem increases. All the experiments which are going to be presented in
chapter 4, are in two dimensions.
The other important thing to measure is the cost in terms of work units (WU).

CHAPTER 3. MULTIGRID METHODS

28

Here a work unit is dened to be the cost of performing one relaxation sweep on
the nest grid. Equivalently, it is the cost of expressing the ne grid operator.
It is customary to neglect the cost of intergrid transfer operations which could
amount to 15 ; 20% of the cost of the entire cycle. First consider a V-cycle with
one relaxation sweep on each level (s1 = s2 = 1). Each level is visited twice and
the grid  requires s; work units. Adding these costs and again using the
sh

d

geometric series for an upper bound give
MV computation cost =
2f1 + 2; + 2;2 + : : : + 2; g < 1 ;22; WU
d

d

nd

d

3.3.2 Convergence analysis

We can attempt to give heuristic and qualitative arguments suggesting that standard multigrid schemes, when applied to well-behaved problems (for example,
symmetric, positive denite systems), not only work, but they work very eciently.
We begin with the heuristic argument that captures the spirit of rigorous
convergence proofs. Let us denote the original continuous problem (for example,
one of our model boundary value problems) Ax = b. The associated discrete
problem on the ne grid  will be denoted A x = b . As before, we let v be
h

h

h

h

h

an approximation to x on  . The global error is dened by
h

h

E = x(x ) ; x  1  i j  N ; 1
h

i
j

h

i
j

i
j

and is related to the truncation error of the discretisation process. In general,
the global error may be bounded in norm in the form

jjE jj  Kh  K 2 IR:
h

p

CHAPTER 3. MULTIGRID METHODS

29

The quantity that we have been calling the error,

eh

=

xh

; v , will now
h

be called the algebraic error to avoid confusion with the global error. The algebraic error measures how well our approximations (generated by relaxation or
multigrid) agree with the exact discrete solution.
The purpose of a typical calculation is to produce an approximation v which
h

agrees with the exact solution of the continuous problem. Let us specify a tolerance  with a condition such as

jjx ; v jj < 
h

where x = (x(x1 1) : : :  x(x



N

;1 ;1))

N

T

is the vector of exact solution values. This

condition can be satised if we guarantee that

jjE jj + jje jj < 
h

h

(since jjx ; v jj  jjx ; x jj + jjx ; v jj = jjE jj + jje jj < ).
h

h

h

h

h

One way to ensure that jjE jj + jje jj
h

h

h

is to require jjE jj

< 

h

< =2

and

jje jj < =2 independently. The rst condition determines the grid spacing on the
h

nest grid. It says that we must choose
h < h =


2K

!1

=p

The second condition is determined by how quickly our approximations v conh

verge to x . If relaxation or multigrid cycles have been performed until condition
h

jje jj < =2 is met on grid  , where h < h, then we have converged to the level
h

of truncation.

h

In summary, the global error determines the critical grid spac-

ing h. The only computational requirement is that we converge to the level of
truncation error on a grid with h < h.

CHAPTER 3. MULTIGRID METHODS

30

We now consider the V-cycle scheme applied to a d-dimensional problem with
Nd

unknowns and h = 1=N . We have to assume (and can generally show rigor-

ously) that with xed cycling parameters s1 and s2, the V-cycle scheme has a convergence rate,  , which is independent of h. This V-cycle scheme must reduce the
algebraic error from O(1) (the error in arbitrary initial guess) to O(hp) = O(N ;p )
( the order of the global error). Therefore, the number of V-cycles required, v,
must satisfy  v = O(N p ) or v = O(logN ). This is comparable to the computational cost of the best fast direct solvers when applied to the model problems as
mentioned in chapter 1.
3.4

The Eectiveness of the Relaxation Methods

The main concern of this project is to show the robustness of multigrid methods
numerically, and also to assess whether one relaxation method is more eective
and ecient than another if it is used within multigrid. In this section we would
like to compare the smoothing potential of the damped Jacobi and the GaussSeidel methods using a local mode analysis, see Chan 15].
This local mode analysis (originally devised by Brandt, see 14]), is a general
and eective tool for analysing and predicting the performance of a smoother.
The approach is based on the fact that relaxation is typically a local process
in which information propagates by a few mesh-sizes per sweep. Therefore, one
can assume the problem to be in an unbounded domain, with constant (frozen)
coecients, in which case the algebraic error can be expanded in terms of a

CHAPTER 3. MULTIGRID METHODS

31

Fourier series. To do this, let us rst recall the discrete Fourier theorem.

Theorem 3.4.1
ng

Every discrete function

u : I ! IR I = f(i j ) : 0  i j 
n

n

, can be written as

u =
i
j

Xc


i
j

n

() () = e (

i i1

i
j

+j2 )

p

 i = ;1  = (1 2)



where

c = (n +1 1)2
(


X

u (; )
k
l

) n

k
l

k
l I

and

 = f n2+ 1 (k l) ;m  k l  m + pg
where p = 1 m = (n + 1)=2 for odd n and p = 0 m = n=2 + 1 for even n.
n

Consider the pure-diusion of the form ;r2u = f with homogeneous Dirichlet condition on a unit square discretised using standard second order dierence
approximations with uniform grids. The discretised equation can be expressed as
4u

i
j

4u

i
j

2
= (u +1 + u ;1 + u +1 + u ;1) + k f  1  i j  n ; 1 (3.5)
2
= 4u ; !(4u ; ( k f + u +1 + u ;1 + u +1 + u ;1)): (3.6)
i


j

i

i
j


j

i
j

i
j

i
j

i
j

i
j

i


j

i


j

i
j

i
j

The damped Jacobi iteration corresponding to the equation can be expressed as
2
4^u = 4u ; !(4u ; ( k f + u +1 + u ;1 + u
i
j

i
j

i
j

i
j

i


j

i


j

i
j

+1

+u

i
j

;1 ))

(3.7)

where u^ denotes the new value of u while u denotes the old value of u. The
i
j

i
j

Gauss-Seidel iteration (with lexigraphical order on nodal points, from left to right
and bottom to top) is presented of the form
4^u = (u +1 + u^ ;1 + u
i
j

i


j

i


j

i
j

+1

+ u^

i
j

;1 ) +

k2 f :

i
j

(3.8)

CHAPTER 3. MULTIGRID METHODS

32

We use the discrete Fourier transform to analyse the damped Jacobi method.
Let the global error of the method be dened as ^ = u ;u^ and  = u ;u .
i
j

i
j

i
j

i
j

i
j

i
j

Subtracting the equation (3.6) and (3.7) we obtain
^ =  ; !4 (4 ; ( +1 +  ;1 + 
i
j

i
j

i
j

i


j

i


j

i
j

+1

+

i
j

;1 )):

(3.9)

We write
^ =

X

i
j

c^ () and  =


n

i
j

X

i
j



c ( ):


n

(3.10)

i
j



Substituting the equation (3.10) into (3.9) we have

X
!
c () ; 4c () ; (c +1 ()
4
n
n

+c ;1 () + c +1() + c ;1()) 

!
c^ () = c () ; 4 () ; ( +1 ()
4

+ ;1 () + +1() + ;1()) 

X

c^ () =


i
j







i
j

i
j



i


j







i
j

i


j



i
j



i

i
j



i
j


j

i

i
j

i
j

(3.11)


j

i
j

and comparing the coecient of each (), we obtain that

(3.12)

i
j

c^
c





!

= 1 ; 4 4 ; (e

i1



+ e;i 1 + ei 2 + e;i 2 ) :






(3.13)

Applying the expression of the form e k = cos( )  isin( ) into (3.13) gives
i

() = 1 ; ! (1 ;

k

k

cos1 + cos2

2

)

(3.14)

where
() 
^ is called the amplication factor of the local mode ().
c

i
j

c

The smoothing factor introduced by Brandt is the following quantity
 = supfj
()j =2  j

k

j  

k = 1 2g:

Roughly speaking, the smoothing factor  is the maximal amplication factor
corresponding to those high frequency local modes that oscillate within a 2h
range (and hence can not be resolved by the coarse grid of size 2h).

CHAPTER 3. MULTIGRID METHODS

33

For damped Jacobi method, it is easy to see that
 = maxfj1 ; 2! j j1 ; ! j j1 ; 3!=2jg:

The optimal ! that minimises the smoothing factor is
! = 4=5  = 3=5

(this value of ! is used in the numerical experiments presented in chapter 4). For
! = 1 we have  = 1. This means that the undamped Jacobi method for this

model problem, although convergent as an iterative method by itself, should not
be used as smoother.
We next examine the smoothing potential of the Gauss-Seidel iteration. Unlike the Jacobi method, Gauss-Seidel method depends on the ordering of the
unknown. The most natural ordering is perhaps the lexicographic order. Hence,
we also express the Gauss-Seidel method in terms of the global error denition
by subtracting the equation (3.5) and (3.8). Thus, the method reads
1
4

^ = ( +1 + ^ ;1 + 
i
j

i


j

i


j

i
j

+1

+ ^

i
j

;1 ):

Again using the Fourier transform (3.10), we obtain that
X 1
c +1 () + c^ ;1 ()
c^ () =
n 4
n

+c +1() + c^ ;1() 

c^
c^
1
+1 () + ;1 () + +1 ()
() =
c
4
c

c^
+ c ;1 (  ) 
c^
+1 () + +1()
=
c
4 (  ) ; ;1 (  ) ; ;1 (  ) :

X







i
j

i


j



i


j





i
j





i
j

(3.15)



i

i
j


j

i




j

i
j





i
j







i

i
j


j

i
j

i


j

i
j

(3.16)
(3.17)

CHAPTER 3. MULTIGRID METHODS
Employing the expression () = e (

i i1

i
j

c^
c

+j2 )

34
we have

ei((i+1)1+j2 ) + ei(i1 +(j +1)2 )
:
i i1 +j2 )
ei((i;1)1 +j2 ) ei(i1 +(j ;1)2 )

= 4e (

;

;

(3.18)

Finally, reducing the exponential term we obtain the local amplication factor as
follows:
()

ei1 + ei2
:
e;i1 e;i2

= 4;

;

(3.19)

It is elementary to determine that

j

j = 1=2:

 =
(=2 cos;1 (4=5))

We can see that the smoothing factor of the Gauss-Seidel is less than the damped
Jacobi, this means the Gauss-Seidel is slightly better smoother than the damped
Jacobi method.

Chapter 4
Numerical Results
We are now in a position to test the relaxation and multigrid methods in a
practical application. Numerical results based on the theory discussed in chapters
1{3 are presented here.
The discretisation process for some model diusion and convection-diusion
problems will be accomplished using some freely available Matlab software for
solving such problems, written by D.J. Silvester (available from http://www.ma.
). The relaxation and multigrid methods were also

umist.ac.uk/djs/index.htm

written as Matlab programs and these are given in appendix A.
We use the V{Cycle algorithm as a representative multigrid method, and
perform one relaxation sweep on each level (s1=s2=1). We will discuss the numerical results separately for diusion and convection-diusion, and give some
discussion of the V{Cycle method robustness in each case.

35

CHAPTER 4. NUMERICAL RESULTS

36

4.1 Numerical Solution of the Diusion Problem
The general test problem in chapter two is ;r2u + w  ru = f . If we set w=0
the problem is ;r2u = f , and it is then referred to as the diusion problem.
With the available software we can generate the linear system of equations using
uniform grids of square elements with  = 1, f = 0. The solution satises the
condition u = 1 on part of the bottom boundary and the right-hand wall, and
u

= 0 on the remainder of the boundary.
Applying the V-Cycle method, the solution of our tes