B ¦ 1‘ ( &

¦

V

C 0

and substituting in (2.32) gives the linear problem

p… 1‘ ( &… Y

t

¦ ¨ ¢ ¢ &¥

0

where

C y¦ y¡ B g C z¦ Ty¦ B ‚ XH…x

… ¦

The above equations form a linear system of equations

‚

¦

H…#

… ¦

in which the coef¬cients of are the ™s; those of are the ™s. In addition, is a

Symmetric Positive De¬nite matrix. Indeed, it is clear that

£¢

£¢

x 9 y¦ z¦ ‚ 9z¦ y¦

… … ¦ ¦

d u„

u n £ § ©

¢ § ¥ © U© © § ¥ £ © ¡ §

¥ © ¡ £

¥ ¡

(C ¦ B ‚

T¦ … x ¦ B ‚ H… 7

which means that . To see that is positive de¬nite, ¬rst note that V V

C

¨

7 7

for any function . If for a function in , then it must be true that

V

¦ £

almost everywhere in . Since is linear in each triangle and continuous, then it is clear

£

that it must be constant on all . Since, in addition, it vanishes on the boundary, then it

£

must be equal to zero on all of . The result follows by exploiting the relation

¦ 1‘ ( & ’¦ ¦ x¦ B u B

‚ C

with

C

0

1( %

0

! £ " %%%$

$$$

which is valid for any vector .

z¦ h¦ ‘ H…v

Another important observation is that the matrix is also sparse. Indeed, is

…

nonzero only when the two basis functions and have common support triangles,

H…

or equivalently when the nodes and are the vertices of a common triangle. Speci¬cally,

for a given node , the coef¬cient will be nonzero only when the node is one of the

nodes of a triangle that is adjacent to node .

In practice, the matrix is built by summing up the contributions of all triangles by

applying the formula

&

C z¦ Cy¦ B ‚

… z¦ Cy¦ B ¥ ‚

…C

¥

in which the sum is over all the triangles and

x 2u¦ y¦ ¥ ¢ C z¦ Cy¦ B ¥ ‚

…

9… ¦

z¦ Ty¦ B ¥ ‚

…C

Note that is zero unless the nodes and are both vertices of . Thus, a triangle

w

contributes nonzero values to its three vertices from the above formula. The matrix

¦ ‘z¦ B ¥ ‚ C z¦¦ ‘C“¦¦ BB ¥¥ ‚‚ C y¦¦ ‘C“¦¦ BB ¥¥ ‚‚ 3 ¥

u z “ z …

…… …

8 ' ¦ Cy¦ B ¥ ‚

A C' …

5

C ' ¦ ' ¦ B ¥ ‚ C z ¦ ' ¦ B ¥ ‚ C y ¦ ' ¦ B ¥ ‚

…C C

C

‘r`rB

Ya`

associated with the triangle with vertices is called an element stiffness

C

matrix. In order to form the matrix , it is necessary to sum up all the contributions

¦ ' ¦ B ¥ ‚

to the position of the matrix. This process is called an assembly pro-

H”

C

cess. In the assembly, the matrix is computed as

!

1( ‘ & Y

t

£¢

¡

¨ £ ¢ &¥

0

£¢

Q˜

¤

in which is the number of elements. Each of the matrices is of the form