Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 151

5.4 Lemma. The mapping

0

div : RT 0 → M

is surjective.
Proof. After enlarging by finitely many triangles if necessary, we may assume
that is convex. Given f ∈ M

0 , there is a u ∈ H 2 () ∩ H 0 () such that

u = f . Set q :

=

grad u . By Gauss' integral formula we have

q · nds = div qdx =

fdx.

∂T T T

From (5.9a) we conclude that T div ρ qdx = ∂T (ρ q) · nds = T fdx. Since
div ρ q and f are constant in T , it follows that div ρ q = f .

0

→ RT 0 in the construction above is
bounded. Therefore, recalling Fortin's criterion we see that the inf-sup condition
has been established simultaneously.
The error of the finite element solution will be derived from the approximation
error.

Finally we note that the mapping M

5.5 Lemma. Let T h be a shape-regular triangulation of . Then

q − ρ q H( div ,) ≤ ch | q | 1 + inf

v h ∈ M

0

div q − v h 0 .

Proof. We first consider the interpolation on a triangle. By the trace theorem the

functional q → e q · nds , e ⊂ ∂T , is continuous on H 1 (T ) 2 . Moreover, we have
ρ T q = q for q ∈ P

2 2

0

0 since

P ⊂ RT 0 . Therefore, the Bramble-Hilbert lemma and

a scaling argument yield

q − ρ q 0 ≤ ch | q | 1 .

The bound for div (q − ρ q) follows from the minimal property 5.3, and the proof
is complete.
Now the error estimate of the finite element solution of (5.1)

σ − σ h H( div ,) + u − u h 0
≤ c(h | σ | 1 + h u 1 +

( 5 . 13 )

0 f − f h 0 )

inf
f h ∈ M

is a direct consequence of Theorem 4.5.
Moreover, there is a comparison with the standard finite element approxima-

tion.