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216 Polynomial Approximation of Differential Equations
(9.8.3) −[F(x,y1) − F(x,y2)](y1 − y2) ≥ ǫ (y1 − y2) 2 , ∀x ∈ [−1,1], ∀y1,y2 ∈ R.
For a regular F this is equivalent to requiring − ∂F
∂y
≥ ǫ in [−1,1] × R.
It is clear that (9.8.2) is not equivalent to a linear system and the unknown vector
¯v := {pn(η (n)
i )}0≤i≤n has to be computed by an iterative approach. We can adopt the
Richardson method (see section 7.6) by defining the sequence ¯v (k) ≡ (v (k)
0 , · · · ,v(k) n ),
k ∈ N, such that
(9.8.4) ¯v (k+1)
:= (I − θD)¯v (k) + θ ¯w (k) , k ∈ N,
where ¯v (0) is the initial guess. In (9.8.4), D is the matrix corresponding to the system
(7.4.1) and ¯w (k) ≡ σ,F(η (n)
1 ,v (k)
1
), · · · ,F(η(n) n ,v (k)
n ) , k ∈ N. If the iterates converge
for some θ > 0, then one gets limk→+∞ v (k)
0 = σ, limk→+∞ v (k) (n)
i = pn(η i ), 1 ≤ i ≤ n.
A faster convergence is realized by using preconditioners (see sections 8.3 and 8.5).
Second-order nonlinear problems can be also considered. An example is given by
the equation
(9.8.5)
⎧
⎨
−U ′′ (x) + F(x,U(x),U ′ (x)) = 0 x ∈] − 1,1[,
⎩
U(−1) = σ1, U(1) = σ2,
where F : [−1,1] × R 2 → R, σ1 ∈ R, σ2 ∈ R, are given.
Spectral type approximations of (9.8.5) are studied in maday and quarteroni(1982)
in the case F(x,y,z) := 1
ǫ yz − f(x), σ1 = σ2 = 0, where ǫ > 0 and the function
f : [−1,1] → R is given. A general trick to analyze the solution of problem (9.8.5) and
its approximation is to write the equation in the form U = LF(x,U,U ′ ), where L is
the linear operator which associates to any function f the solution of problem (9.1.4)
with σ1 = σ2 = 0. Then, the investigation proceeds by using fixed-point theorems and
the compactness of the operator L. More details can be found in brezzi, rappaz and
raviart(1980). We further examine the above example in section 10.4.