Using Triangle Inequality

Now on the other hand, using the triangle inequality and then (33) we have
∥u − uh∥
L1(Ω)
≥ ∥uI − uh∥
L1(Ω)
− ∥u − uI∥
L1(Ω)
≥ ∥uI − Tpuh∥
L1(Ω)
− ∥uh − Tpuh∥
L1(Ω)
− ∥u − uI∥
L1(Ω)
≥ ∥uI − Tpuh∥
L1(Ω)
− ∥uh − Tpuh∥
L1(Ω)
− C2hp+1|u|
Wp+1
1 (Ω);
≥ ∥uI − Tpuh∥
L1(Ω)
− C∥u − uh∥
L1(Ω)
−
−C3hp+1|u|
Wp+1
1 (Ω); from which we conclude that
∥u − uh∥
L1(Ω)
≥ C0∥uI − Tpuh∥
L1(Ω)
− C3hp+1|u|
Wp+1
1 (Ω)
≥ C0 vuut ΣN i=1 ∥uI − Tpuh∥2
L1(Ω∗
i )
− C3hp+1|u|
Wp+1
1 (Ω)(36) ; where Ω∗ i is as in the proof of Theorem 2.4. By a standard scaling argument, and
(30) or the argument leading to it, we have
∥uI−Tpuh∥
L1(Ω∗ i )
≥ C2hmin∥uI−Tpuh∥
L1(Ω∗
i )
≥ C2h2 minhp+1 √
Q(˜D0
i ; ˜D1 i ; · · · ; ˜Dp i ):
This completes the proof. 
Theorem 2.7. Suppose
H∞ : u ∈ …show more content…
As before, for u ∈ Wp+1
∞ (Ω), there is a uI ∈ P∗ p;h so that
∥u − uI∥
L∞(Ω)
≤ Chp+1|u|
Wp+1
∞ (Ω); where C > 0 is a constant independent of u and h.
We proceed exactly as in Theorem 2.4. Let xi be a midpoint of an edge in , and recall that Tpuh is the Taylor polynomial of degree p evaluated at xi over the triangle i. Then
(38) ∥uh − Tpuh∥
L∞(i)
≤ Chp+1
i
|uh|
Wp+1
∞ (i)
≤ Chp+1

|uh|
Wp+1
∞ ():
Applying Lemma 2.3 with m = p + 1, s = ∞ and v = uh and using the regularity condition (17) and equivalence of norms on ˆ, we have
|uh|
Wp+1
∞ ()
≤ C
−(p+1)

∥uh∥
L∞(ˆ) so that by (17) hp+1 
|uh|
Wp+1
∞ ()
≤ C∥uh∥
L∞():
Taking the maximum of (38) over , we have, using the last inequality,
(39) ∥uh − Tpuh∥
L∞(Ω)
≤ C∥uh∥
L∞(Ω):
OPTIMAL ORDER CONVERGENCE IMPLIES NUMERICAL SMOOTHNESS 11
So as before using the stability of Tp in the L∞ norm and the approximation property of Ph [5], we conclude that
∥u − uh∥
L∞(Ω)
≥ C0 max i ∥uI − Tpuh∥
L∞(Ω∗
i )
− Chp+1|u|
Wp+1
∞ (Ω); where Ω∗ i is as in the proof of Theorem 2.4. Let Ui = ∥uI − Tpuh∥
L∞(Ω∗
i ) and use
(30) to derive
U2
i =
1
|Ω∗ i |
∫
Ω∗ i U2 i dx ≥ 1
|Ω∗
i
|
∫
Ω∗
i
(uI − Tpuh)2dx
≥ Ch
−2∥uI − Tpuh∥2
L2(Ω∗
i )
≥ …show more content…
Suppose ∥u − uh∥
Ls(Ω)
≤ Chp+1+;  ≥ 0. Applying this to inequality (40) deduces the result. Other assertions follow in a similar way. 
Note that all D i need to be bounded for convergence as a consequence of this theorem. Theorem 3.2. Suppose that u ∈ Cp+1(Ω); s = 1; 2;∞;Ω ⊂ R2 and that uh ∈
Wh on a quasi-uniform family {Qh} of meshes on Ω into quadrilaterals. Then a necessary condition for
∥u − uh∥
Ls(Ω) = O(hp+1) is for uh to be Wp+1 s smooth. In particular, for
∥u − uh∥
L∞(Ω) = O(hp+1) a necessary condition is that all the kth partial derivatives at xi ∈ T satisfy
(41) @(u − uh)(xi) = O(hp+1−k); || = k; 0 ≤ k ≤ p:
In other words, we have a simultaneous approximation result. Here all smoothness refers to interior smoothness and {xi} is any collection of points, one from each element. Proof. Let Qh be a quasi-uniform subdivision on Ω in R2, and let u ∈ Wp+1
∞ (Ω) and uI ∈ Php be such that
∥u − uI∥
L∞(Ω)
≤ Chp+1|u|
Wp+1
∞ (Ω):
Let uh ∈ Qhp be given and to simplify the presentation, we will use shorthand notations: let || = k and since we will treat one kth derivative at a time, there is no ambiguity in setting u(k) h = @uh, u(k)
I = @uI , and u(k) = @u. At a typical point xm ∈  ∈ Qh, we denote by Tpuh the Taylor polynomial of degree p