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11
DOMAIN-DECOMPOSITION
METHODS
For several reasons, when approximating differential equations numerically, it is often
convenient to decompose the domain, where the solution is defined, into different sub-
sets. Then, independently in each subdomain, one computes polynomial approximations
by the techniques introduced in the previous chapters. The goal now is to find a suitable
way to match the different pieces to obtain a discretization of the global solution.
11.1 Introductory remarks
Recently, domain-decomposition (or multidomain) methods have become a fundamental
subject of research in spectral methods. This is especially true for boundary-value
problems in two or more space variables, when the solutions are defined in domains
with a complicated geometry. Actually, spectral-type techniques are well suited for very
simple domains, obtainable by cartesian products of intervals (see chapter thirteen). It
is clear that, when the given domain does not conform with these requirements, setting
up a spectral approximation scheme is not a straightforward procedure. From this point
of view, this is a severe drawback in comparison with other more flexible methods, such
as finite-differences or finite element methods.