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 methods from a domain decomposition perspective. An important feature of this approach is the systematic use of “fully discontinuous functions” and the treatment of a general boundary value problem with prescribed jumps. Usually finite element methods are developed using splines, but a more general point of view is obtained when they are formulated in spaces in which the functions together with their derivatives may have jump discontinuities and in the general context of boundary value problems with prescribed jumps. Two broad classes of Trefftz methods are obtained: direct (Trefftz—Jirousek) and indirect (Trefftz—Herrera) methods. In turn, each one of them can be divided into overlapping and nonoverlapping. The generality of the resulting theory is remarkable, because it is applicable to any partial (or ordinary) differential equation or system of such equations, which is linear. The article is dedicated to Professor Jiroslav Jirousek, who has been a very important driving force in the modern development of Trefftz method.