APPROXIMATE METHODS 3

and,

βι =

u — ü

Φ 0 on /\

ε

2

= q

—

q Φ 0 oni

2

Our aim is now to make this error as small as possible over the

domain and on the boundary. In order to do so the errors can be

distributed and the way in which this distribution is carried out

produces different types of weighted residual techniques.

1.2 WEIGHTED RESIDUAL TECHNIQUES

The simplest of these techniques starts by exactly satisfying the

boundary conditions (i.e. ε

ι

= ε

2

= 0) and distributing the error

according with a weighting function w. This function is such that it

identically satisfies the homogeneous boundary conditions and can be

written as,

>ν

= 0

1

ΐΑΐ+02</'2+03</'3+ ..· (1.8)

The

ij/i

are members of a set of known linearly independent functions

and ßi are arbitrary coefficients. We will call them w

t

when they are

arbitrary nodal coefficients.

The distribution of the error function ε can now be carried out by

multiplying it by the weighting function w and integrating over the

domain, i.e.

ewdQ= (V

2

u)wdQ = 0 (1.9)

In this way the error is distributed in accordance with the functions in

w. Note that because the

/?,

coefficients are arbitrary, (1.9) can also be

written as,

(V

2

i#

t

dß = 0 for i = 1, 2, . . ., n (1.10)

h

Classical finite difference techniques for instance can be interpreted

as a special case of equation (1.9) for which the weighting functions

are Dirac delta functions, and the ß

t

are nodal arbitrary coefficients

(w,),

such that,

w = w

l

6

1

4-νν

2

^

2

+

νν

3^3 + .·. (1-11)

4 APPROXIMATE METHODS

The Dirac delta function is a generalised function which can be

defined as the limit of a normal function such as

■im

Sin[

*

(

*-*'

)]

=*(*-*,) (1.12)

yv-oo n(x-Xi)

In the limit this function is zero at every point except where the

argument is zero where it is infinite (i.e. at x = x,). Thus it represents a

point singularity at the 'source' point x

t

.

The delta function has the property that

Γ °° CXi + a

ö(x-x

i

)dx=\ (5(x-x;)dx=l (1.13)

J -

oo

Jx, - a

for a any positive number.

We also have that for any function/(x) (continuous at x

f

) we can

write

r oo ΛΧ,- + a

f(x)S(x-

Xi

)dx= f(x)ö(x-x

i

)dx=f(x

i

) (1.14)

J -

oo

Jx,-ü

Where it is not necessary to represent the arguments of the functions

explicitly in the analysis we shall use the shorthand notation putting

ö(x-x

i

) = ö

i

and (1.15)

f(Xi)=fi

for simplicity. Where it is important to include the arguments for

clarity of explanation we shall use the full notation writing x and x,

explicitly.

The approximating functions are given by

u = η

1

φ

ί

+ η

2

φ

2

+ ιι

3

φ

3

+ . . . (1.16)

(The φί are localised second-order polynomials taken over the

subdomain surrounding the node for the central difference scheme.)

Substituting (1.16) into (1.9) produces the following equation at

each node T:

e<

5

i

.dO =

ei

= 0 (1.17)

Ω

The above presentation gives the same numerical results as the

central finite difference scheme for quadratic approximating func-

tions.

The procedure is however, more general as it allows us equally

APPROXIMATE METHODS 5

well to use other types of φ

{

functions. Furthermore the basic ideas

can be generalised. Using curvilinear coordinates these generali-

sations give rise to the 'cell collocation' method.

1

Basically this is a

subregion collocation technique as shown in Example 1.1.

Example 1.1

Consider the equation

dx

where x varies from 0 to 1 with boundary conditions

u(0)

= u(l) = 0 (b)

Cell

2

+

w

+ x = 0 xe[0, 1] (a)

Figure 1.1 Point under consideration * +

>

%—

/-1 / /♦!

* ; *

We can now propose over each subregion or 'cell' (Figure 1.1) the

following approximating function:

" = "i-101+"i02 + "i+103 (C)

where,

φ

2

= (1-ξ)(ί + ξ) (d)

ζ is a dimensionless coordinate, ξ = x/\L If

we

differentiate equation

(a) with functions (d) one finds the following algebraic relationship:

(d

2

u

/4 8 4

= \j2

U

i-l--J2

U

i + ]2

U

i+l

J

+

U

i +

X

i

4

= 72("i-i-

2

»i +

M

i+i) +

M

i + -

x

i (e)

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