The purpose of this text is to offer a comprehensive and self-contained
presentation of some of the most successful and popular domain decomposition
methods for partial differential equations. Strong emphasis is put on both
algorithmic and mathematical aspects. In addition, we have wished to present
a number of methods that have not been treated previously in other
monographs and surveys. We believe that this monograph will offer something new
and that it will complement those of Smith, Bj0rstad, and Gropp [424] and
Quarteroni and Valli [392]. Our monograph is also more extensive and broader
than the surveys given in Chan and Mathew [132], Farhat and Roux [201], Le
Tallec [308], the habilitation thesis by Wohlmuth [469], and the well-known
SIAM Review articles by Xu [472] and Xu and Zou [476].
Domain decomposition generally refers to the splitting of a partial
differential equation, or an approximation thereof, into coupled problems on smaller
subdomains forming a partition of the original domain. This decomposition
may enter at the continuous level, where different physical models may be
used in different regions, or at the discretization level, where it may be
convenient to employ different approximation methods in different regions, or in
the solution of the algebraic systems arising from the approximation of the
partial differential equation. These three aspects are very often interconnected
in practice.