6¥

all functions whose derivatives up to the ¬rst order are in . This is known as .

C

However, this space does not take into account the boundary conditions. The functions in

£ B 0&

¨ must be restricted to have zero values on . The resulting space is called . ¥ C

The ¬nite element method consists of approximating the weak problem by a ¬nite-

¨

dimensional problem obtained by replacing with a subspace of functions that are de¬ned

as low-degree polynomials on small pieces (elements) of the original domain.

¡ T ¥ ¢

D ¡ Finite element triangulation of a domain.

£

Consider a region in the plane which is triangulated as shown in Figure 2.7. In this

example, the domain is simply an ellipse but the external enclosing curve is not shown.

£

The original domain is thus approximated by the union of triangles ,

1”( £ ¡

0

For the triangulation to be valid, these triangles must have no vertex that lies on the edge

” ” ”n

uu u ™

n

¢

£ ¤§ ¥ §¢

¡

¡© £ £ ¥ £ ¤ § ¨¡ ¢

§¥ © ©

of any other triangle. The mesh size is de¬ned by

§¦¥0¤ ¢ ( C

B

diam

" %%%"

$$$

”

B

where diam , the diameter of a triangle , is the length of its longest side.

¨

C

Then the ¬nite dimensional space is de¬ned as the space of all functions which

£

are piecewise linear and continuous on the polygonal region , and which vanish on the

£ ¥

boundary . More speci¬cally, ¥

¢ ¡¦ ¦ ¨

z ¢ £ ¦

b

¦

§¥ ¤

¦ !

78

continuous linear

˜ i1)1Y ‚r‘…y 0 ¨ …y¦¦

¦

£

¦

Here, represents the restriction of the function to the subset . If

¤

are the nodes of the triangulation, then a function in can be associated with each

z

… “¦

…

¦

node , so that the family of functions ™s satis¬es the following conditions:

…… £ H… C B … ¦

¨ ¦ ¢Y ¥

t ¦ ¦

if

a

&

¦

7 ¦ ¦

if

˜ 111)‘ dTPd¦

d¦

These conditions de¬ne uniquely. In addition, the ™s form a basis of the

¨

space .

¨

Each function of can be expressed as

C B y¦ 1 01‘ ( & C B ¦

"

¦ ¦

The ¬nite element approximation consists of writing the Galerkin condition (2.30) for func-

¨

tions in . This de¬nes the approximate problem:

Y

t

¨ C y¡ B C B ‚

¨ ¨ & ¢ &¥

Find such that

V V

‚ # ¦ y¡ B ¦ B ‚

¨

˜

Since is in , there are degrees of freedom. By the linearity of with respect to , it

V

˜ i1)1Y

is only necessary to impose the condition for . This results

V

C

C

˜

in constraints.

d¦

!