Trefftz method: A general theory

A precise definition of Trefftz method is proposed and, starting with it, a general theory is briefly explained. This leads to formulating numerical m ethods from a domain decomposition perspective. An important feature of thi s approach is the systematic use of "fully discontinuous functions" and the treatment of a general boundary value problem with prescribed jumps. Usual ly finite element methods are developed using splines, but a more general p oint of view is obtained when they are formulated in spaces in which the fu nctions together with their derivatives may have jump discontinuities and i n the general context of boundary value problems with prescribed jumps. Two broad classes of Trefftz methods are obtained: direct (Trefftz-Jirousek) a nd indirect (Trefftz-Herrera) methods. In turn, each one of them can be div ided into overlapping and nonoverlapping. The generality of the resulting t heory is remarkable, because it is applicable to any partial (or ordinary) differential equation or system of such equations, which is linear. The art icle is dedicated to Professor Jiroslav Jirousek, who has been a very impor tant driving force in the modern development of Trefftz method. (C) 2000 Jo hn Wiley & Sons, Inc.