Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 185

Proof. For convenience, we restrict ourselves to the case f = f h since the effect
of the data oscillation is absorbed by the second terms in the inequalities. Only the
residuals R T and R e defined in (8.2) and (8.3) enter into Algorithm 9.3. Therefore,
the normal components of σ ω z on the edges are bounded,
| σ ω z · n |≤ c h

.

e ⊂ ω z | R e |

T ⊂ ω z | R T |+

Since the broken Raviart-Thomas functions are piecewise polynomials with a fixed
number of degrees of freedom, we have
σ ω z 0 ,T ≤ ch T

e ⊂ ω z | σ ω z · n |
with c being a constant that depends only on the shape parameter. Hence,
σ ω z ≤ c

T

2
0

η T,R

⊂

ω z

and another summation yields

σ 0 ≤ cη R .

This proves (9.9). The efficiency of the residual estimators (8.13) implies (9.10).

As a byproduct we obtain the comparison of the performance of the P 1 ele-
ment and of the mixed method with the Raviart-Thomas element that was stated
in §5.
Proof of Theorem 5.6. We conclude from the preceding discussion that
∇ u − σ h

2
0

2
0

σ

2
0

+ ch 2

2
0

+∇ (u − u h ) ≤ 2 f − f h

2
0

ch 2

2

≤ c ∇ (u − u h ) + f − f h

0 ,

and the proof is complete.

Problem

9.6 Consider the Helmholtz equation
− u + αu = f

in ,

0 n ∂
with α> 0. Let v ∈ H 0 () and σ ∈ H( div ,) satisfy div σ + f = αv .
Show the inequality of Prager-Synge type with a computable bound
| u − v |

u =

2
1

2
0

+ α u − v

( 9 . 11 )

2
0

2
0

2

+

grad u − σ

+ α u − v =

grad v − σ

0 .

Recall the energy norm for the Helmholtz equation in order to interpret (9.11).