5 . The Indirect Approach To Domain Decomposition

1. Introduction. The main objective of DDM is, when a domain Ω and one of its partitions are given, to obtain the solution of a boundary value problem defined on it (the 'global problem'), by solving problems formulated on the subdomains of the partition (the 'local problems'), exclusively. This objective can be achieved if sufficient information about the global solution is known, on the internal boundary (which separates the subdomains from each other and to be denoted by Σ), for defining well-posed problems in each one of the subdomains of the partition. Herrera proposed recently a general and unifying theory [15],[14], in which DDM are interpreted as methods for gathering such information. According to it, one defines an information-target on Σ, referred as the sought information [15], and the objective of DDM is to obtain such information. There are two main procedures for gathering the sought information, which yield two broad categories of DDM: direct methods and indirect (or Trefftz-Herrera) methods. This paper belongs to a sequence of papers [15],[6],[5], [4],[21], included in this Proceedings, in which an overview of Herrera's unified theory is given. In particular, the present paper is devoted to a systematic presentation of indirect methods, and a companion paper deals with direct methods [6]. introduced indirect methods in numerical analysis. They are based on the Herrera's Algebraic Theory of boundary value problems [9],[10],[8]. Numerical procedures such as Localized Adjoint Methods (LAM) and Eulerian-Lagrangian LAM (ELLAM) are representative applications [17],[3]. A large number of transport problems in several dimensions have been treated using ELLAM [20]. Indirect Methods of domain decomposition stem from the following observation: when the method of weighted residuals is applied, the information about the exact solution that is contained in the approximate one is determined by the family of test functions that is used, exclusively [9],[16],[10]. This opens the possibility of constructing and applying a special kind of weighting functions, which have the property of yielding the sought information at the internal boundary Σ, exclusively, as it is done in Trefftz-Herrera Methods. The construction of such weighting functions requires having available an instrument of analysis of the information supplied by different test functions. The natural framework for such analysis is given by Green's formulas. However, the conventional approach to this matter is not sufficiently informative for applications to domain decomposition methods. Indeed, in the usual approach [19], one considers the Green's formula