Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 163

(i, j + 1 ) (i + 1 ,j + 1 )

• •

(i + 1 / 2 ,j + 1 / 2 )

•

• •

(i, j) (i + 1 ,j)

Fig. 37. Numbering of the nodes in the element T ij for the Q 1 - P 0 element

One indicator of the instability is the fact that the kernel of B h : M h −→ X h
is nontrivial. In order to avoid unnecessary indices when showing this, we will
denote the vector components of v by u and w , i.e.,

u
w

.

v =

With the numbering shown in Fig. 37, the fact that q is constant and div v is linear
implies

q div vdx = h 2 q i + 1 / 2 ,j + 1 / 2 div v i + 1 / 2 ,j + 1 / 2

T ij

1
2 h [ u i + 1 ,j + 1 + u i + 1 ,j − u i,j + 1 − u i,j

= h 2 q i + 1 / 2 ,j + 1 / 2 ( 7 . 1 )

+ w i + 1 ,j + 1 + w i,j + 1 − w i + 1 ,j − w i,j ] .

We now sum over the rectangles. Sorting the terms by grid points is equivalent to
partial summation, and we get

q div vdx = h 2
i,j

[ u ij ( ∇ 1 q) ij + w ij ( ∇ 2 q) ij ] , ( 7 . 2 )

where

1
2 h [ q i + 1 / 2 ,j + 1 / 2

( ∇ 1 q) i,j = + q i + 1 / 2 ,j − 1 / 2 − q i − 1 / 2 ,j + 1 / 2 − q i − 1 / 2 ,j − 1 / 2 ] ,

1
2 h [ q i + 1 / 2 ,j + 1 / 2 + q i − 1 / 2 ,j + 1 / 2 − q i + 1 / 2 ,j − 1 / 2 − q i − 1 / 2 ,j − 1 / 2 ]

( ∇ 2 q) ij =

are the difference quotients. Since v ∈ H 0 () 2 , the summation runs over all
interior nodes. Now q ∈

ker (B h ) provided

q div vdx = 0 for all v ∈ X h ,