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282 Polynomial Approximation of Differential Equations
Numerous theoretical results pertaining to (13.1.1) are available. For a preliminary
introduction, we mention for instance courant and hilbert(1953), sneddon(1957),
sobolev(1964), weinberger(1965). Here, we only state the following theorem.
Theorem 13.1.1 - Let f be a continuous function in Ω satisfying
then there exists a unique solution U ∈ C 0 ( ¯ Ω) ∩ C 1 (Ω) of (13.1.1).
Ω f2 dxdy < +∞,
The aim of the following sections is to suggest a technique for approximating the
solution of Poisson’s equation by means of algebraic polynomials in two variables.
13.2 Approximation by the collocation method
In the following, for any n ∈ N, P ⋆ n denotes the space of polynomials in two variables
which degree is less or equal to n for each variable. Thus, the dimension of the linear
space P ⋆ n is (n + 1) 2 . For n ≥ 2, we also define
(13.2.1) P ⋆,0
n :=
The dimension of P ⋆,0
n is (n − 1) 2 .
p ∈ P ⋆ n
p ≡ 0 on ∂Ω .
We analyze the approximation of problem (13.1.1) by the collocation method. For
any n ≥ 2, let η (n)
j , 0 ≤ j ≤ n, be the nodes in [−1,1] associated to the Ja-
cobi Gauss-Lobatto formula (3.5.1). Then, in ¯ Ω we consider the set of points ℵn :=
0≤i≤n
0≤j≤n
(η (n)
i ,η (n)
j ) . These nodes are displayed in figure 13.2.1 for n = 8 in the case
α = β = − 1
2 . We note that, if a polynomial in P⋆ n vanishes on ℵn ∩ ∂Ω , then it
belongs to P⋆,0 n . Moreover, any polynomial in P⋆,0 n
values attained at the points in ℵn ∩ Ω.
is uniquely determined by the