Untitled - Cdm.unimo.it

Time-Dependent Problems 227
(10.2.18)
Therefore
(10.2.19)
−
1
1
−1
= −ζ
−1
∂V
∂t (χn
− pn) w dx −
1
1
2
−1
∂(χn − pn)
∂x
d
dt χn − pn 2 w,n ≤
∂V
∂t (χn − pn)w dx =
n
j=0
∂[(χn − pn)w]
∂x
n
j=0
∂pn
∂t (χn
− pn) (η (n)
j ,t) ˜w (n)
j
dx ≤ 0, ∀t ∈]0,T].
∂χn
∂t (χn
− pn) (η (n)
j ,t) ˜w (n)
j
( ∂χn
∂t ,χn − pn)w,n − ( ∂χn
∂t ,χn
− pn)w
+ ( ∂(χn − V )
,χn − pn)w, ∀t ∈]0,T].
∂t
The estimate of the error V −pnw ≤ V −χnw+χn−pnw is now a technical matter.
The difference between the inner products (·, ·)w,n and (·, ·)w can be bounded with
the help of (3.8.15) (see also section 9.4). In addition, χn converges to V for n → +∞,
by virtue of (9.4.7). Theorem 10.1.1 is used as done in the proof of stability. The reader
can easily provide the details.
It turns out that, for any t ∈ [0,T], the error decays for n → +∞, with a rate only
depending on the regularity of V . However, using inequality (10.1.2), the multiplicative
term e t appears in the right-hand side of the estimates. Therefore, for large times, the
error bound is not in general very meaningful for practical applications. Refinements
are possible, for example when σ1 ≡ σ2 ≡ 0. In this case, the solution V (·,t) and the
polynomial pn(·,t) are known to decay exponentially to zero for t → +∞.
The convergence analysis of the collocation method for the heat equation is de-
veloped in canuto and quarteroni(1981). The same conclusions hold for the tau
and Galerkin methods. These techniques can be extended to other parabolic operators
or to different boundary conditions, but very few theoretical results are available. For
homogeneous Neumann boundary conditions, an analysis of stability in the Chebyshev
case is provided in gottlieb, hussaini and orszag(1984). A numerical example for
a nonlinear parabolic equation is presented in section 12.1.