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250 Polynomial Approximation of Differential Equations
When possible, the initial domain is divided into subsets, each one having an ele-
mentary shape. Then, the approximation of the global solution results from combining
the different discrete solutions relative to each subdomain. The way of coupling the
various schemes will be discussed in the coming sections.
In this book we treat only one-dimensional problems. In this context, our domains,
being intervals of R, are simple right from the start. Nevertheless, even for equations
in one variable, there are several situations where a domain-decomposition method is
preferable to a direct computation in the whole domain. This is the case for instance
for solutions displaying different regularity behaviors in different parts of the domain.
A global high-degree polynomial approximation is generally more expensive and less
accurate than using high-degree polynomials only in the regions where the degree of
smoothness is lower. In this way, we can treat sharp gradients or shocks when we can
guess their location. Moreover, different techniques can be used in each subdomain
(see the example of section 12.2). The partitioning of the domain is often suggested by
the problem itself, as a result of matching different type of equations. In addition, we
stress that a preliminary theoretical analysis in one variable is often necessary before
experimenting with more complicated situations.
We denote by I an open interval of R. Multidomain methods are usually divided
into two categories: non-overlapping and overlapping. In the first case I is decomposed
into a finite number m ≥ 1 of open intervals Sk, 1 ≤ k ≤ m, in such a way that
Ī =
1≤k≤m ¯ Sk, and Sk1 ∩ Sk2 = ∅, if k1 = k2. In the second case, we assume that
there exist two integers k1 and k2 such that Sk1 ∩ Sk2 = ∅.
11.2 Non-overlapping multidomain methods
Let m ≥ 1 be an integer and sk ∈ R, 0 ≤ k ≤ m, an increasing set of real numbers. We
subdivide the interval I :=]s0,sm[ into the subdomains Sk :=]sk−1,sk[, 1 ≤ k ≤ m.