Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 57

NW N

IV II

III I

W C

E

VI VIII

V VII

S SE

Fig. 10. Numbering of the elements in a neighborhood of the center C and the
neighboring points in the compass directions: E, S, W, N, NW and SE

Table 1. Derivatives of the basis functions ψ C shown in Fig. 10 ( ψ C has the
value 1 at C and is 0 at other nodes)

I II III IV V VI VII VIII

− h − 1 h − 1 h − 1 − h − 1 ∂ 1 ψ C 0 0 0 0

− h − 1 − h − 1 h − 1 h − 1

∂ 2 ψ C 0 0 0 0

We compute the elements of the system matrix A ij , where again we choose
local indices and exploit the symmetry, (see Fig. 11 and Table 1)

( ∇ ψ C ) 2 dxdy a(ψ C ,ψ C ) =

I − VIII

2 [ (∂ 1 ψ C ) 2 + (∂ 2 ψ C ) 2 ] dxdy

=

I + III + IV

2 2 (∂ 1 ψ C ) 2 dxdy + (∂ 2 ψ C ) 2 dxdy

=

I + III I + IV

2 h − 2 2 h − 2 = dxdy + dxdy

I + III I + IV

= 4 ,

a(ψ C ,ψ N ) =

I + IV ∇ ψ C ·∇ ψ N dxdy

( − h − 1 )h − 1 dxdy

= ∂ 2 ψ C ∂ 2 ψ N dxdy =

I + IV I + IV

=− 1 .

Here ψ N is the nodal function for the point north of C. By symmetry, a similar
computation gives

a(ψ C ,ψ E ) = a(ψ C ,ψ S ) = a(ψ C ,ψ W ) = a(ψ C ,ψ N ) =− 1 .