Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 110

in the sum. Thus, t he val ues of the integrals do not change if we subtract the
integral mean value w h (e) on each edge e :

L u (w h ) = ∂ ν u(w h − w h (e))ds.

e

T e ⊂ ∂T

It follows from the definition of w h (e) that e (w h − w h (e))ds =

0. The values of
the integrals also do not change if we subtract an arbitrary constant function from
∂ ν u on each edge e . This can be ∂ ν Iu in particular, and we get

L u (w h ) = ∂ ν (u − I u)(w h − w h (e))ds.

e

T e ⊂ ∂T

It follows from the Cauchy-Schwarz inequality that

2 ds 2 ds 1 / 2

| L u (w h ) |≤

e |∇ (u − Iu) | e | w h − w h (e) |

. ( 1 . 8 )

e ⊂ ∂T

T

We now derive bounds for the integrals in (1.8). By the trace theorem and the
Bramble-Hilbert lemma,

2 ds ≤ c ∇ (v − Iv) 2 2

2 ,T ref ≤ c | v |

2

∂T ref |∇ (v − Iv) |

1 ,T ref ≤ c v − Iv 2 ,T ref ,

for v ∈ H 2 (T ref ) . Using the transformation formulas from Ch. II, §6, we see that

2 ds ≤ ch | v | 2

2 ,T

∂T |∇ (v − Iv) |

( 1 . 9 )

for T ∈ T h . Similarly, for each edge e of ∂T ref ,

2 ds 2

1 ,T ref

c |

2
1 ,T ref

e |

w h − w h (e) | ≤ c w h ≤ w h | for all w h ∈ P 1 .

Here the Bramble-Hilbert lemma applies because the left-hand side vanishes for
constant functions. For e ⊂ T ∈ T h , the transformation theorems yield

2 ds ≤ ch | w h | 2

1 ,T

1
∗

e | w h − w h (e) |

for all w h ∈ M

, 0 .

( 1 . 10 )

We now insert the estimates (1.9) and (1.10) into (1.8), and use the Cauchy-
Schwarz inequality for Euclidean scalar products:

| L u (w h ) |≤ 3 ch | u | | w h |

2 ,T 1 ,T

T

c h
T

1 ,T 1 / 2 2 ,T

T

2 2

≤ | u | | w h |

= c h | u | 2 , w h h .

( 1 . 11 )