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226 Polynomial Approximation of Differential Equations
After integration with respect to the variable t, we have
(10.2.15) pn(·,t) 2 t 1
w,n + 2ζ
0 −1
≤ pn(·,0) 2 w,n +
t
0
∂pn
∂x
∂(pnw)
∂x
dx ds
pn(·,s) 2 w,n + Ĩw,nf(·,s) 2
w,n ds, ∀t ∈]0,T].
The norm ·w,n is defined in (3.8.2) and the interpolation operator Ĩw,n is defined in
section 3.3. Finally, recalling lemma 8.2.1, we can eliminate the integral on the left-hand
side of (10.2.15), since it turns out to be positive. Therefore, theorem 10.1.1 yields
(10.2.16) pn(·,t) 2 w,n ≤ e t
pn(·,0) 2 w,n +
t
0
Ĩw,nf(·,s) 2 w,n e −s
ds ,
∀t ∈]0,T].
Taking into account (10.2.9), one shows that the right-hand side in the above expression
is bounded by a constant which does not depend on n. Thus, (10.2.16) shows that the
norm pn(·,t)w (which is equivalent to the norm pn(·,t)w,n by theorem 3.8.2) is
also bounded independently of n. This is interpreted as a stability condition (compare
with (9.4.3)). Moreover, using again lemma 8.2.1, we can also give a uniform bound
t ∂pn
to the quantity 0 ∂x (·,s)2wds, ∀t ∈]0,T], by estimating the right-hand side of
(10.2.15) with the help of (10.2.16).
The analysis of convergence is similar. For any t ∈]0,T], we define χn(t) :=
ˆΠ 1 0,w,nV (·,t) (see (6.4.13) and (9.4.5)). Thus, by (10.2.12) we can write
(10.2.17)
+
1
−1
1
∂V
∂χn ∂(φw)
φw dx = −ζ
∂t −1 ∂x ∂x
n
(fφ)(η (n)
j=0
j ,t) ˜w (n)
dx
j , ∀φ ∈ P 0 n, ∀t ∈]0,T].
The sum on the right-hand side of (10.2.17) replaces the integral 1
fφwdx, after
−1
noting that f(·,t) ∈ P1, ∀t ∈]0,T]. Subtracting equation (10.2.13) from equation
(10.2.17) and taking φ := χn − pn, we obtain by lemma 8.2.1