Date:

Monday, April 22, 2002

Speaker:

Dr. Ismael Herrera

Affiliation:

Instituto de Geofísica
Universidad Nacional Autonoma de Mexico (UNAM)

Contact Information

iherrera@servidor.unam.mx

Title:

The Indirect Method of Domain Decomposition and the Unified Theory

Abstract:

A unifying formulation of Domain Decomposition Methods (DDM), due to Herrera, has interest because its generality and elegance, and its implications on more specific methodologies of DDM and numerical methods for PDE's, in general. The unifying concept of the theory is that DDM are interpreted as procedures for gathering information on the internal boundaries of the domain partitions, sufficient for defining well-posed local problems. According to it, domain decomposition methods may be classified into two broad categories: direct and indirect (or Trefftz-Herrera methods). This talk is devoted to a brief presentation of the unifying theory and a more detailed one of Trefftz-Herrera methods. The theory is characterized by the systematic use of fully discontinuous functions, both as trial and test functions. Trefftz-Herrera formulation is based on a special kind of Green's formulas applicable to discontinuous functions and one of their essential features is the use of weighting functions which yield information, about the sought solution, at the internal boundary of the domain decomposition, exclusively. Green-Herrera formulas, in turn, are based on an algebraic theory of boundary value problems, previously developed by the author. A special class of Sobolev spaces is introduced in which boundary value problems with prescribed jumps at the internal boundary are formulated. Green-Herrera formulas are applied in such spaces and from them the general framework for indirect methods is derived. Guidelines for the construction of the special kind of test functions are then supplied. The generality of the theory is remarkable, since it is applicable to any linear equation or system of such equations, independently of its type and with possibly discontinuous coefficients. As an illustration, applications to elliptic problems in several dimensions are discussed. For this latter kind of problems, numerical algorithms which possess significant advantages over more standard procedures have been derived.