Computer Assisted Mechanics and Engineering Sciences, 4: 369–382, 1997. Copyright c  1997 by Polska Akademia Nauk Trefftz–Herrera Method Ismael Herrera Instituto de Investigaci´on en Matem´aticas, Aplicadas y Sistemas: IIMAS Universidad Nacional Aut´onoma de M´exico, Apartado Postal 22-582, 14000 M´exico, D.F. M´exico e-mail: iherrera@tonatiuh.igeofcu.unam.mx (Received December 20, 1996) The author’s algebraic theory of boundary value problems has permitted systematizing Trefftz method and expanding its scope. The concept of TH-completeness has played a key role for such developments. This paper is devoted to revise the present state of these matters. Starting from the basic concepts of the algebraic theory, Green–Herrera formulas are presented and Localized Adjoint Method (LAM) derived. Then the classical Trefftz method is shown to be a particular case of LAM. This leads to a natural generalization of Trefftz method and a special class of domain decomposition methods: Trefftz–Herrera domain decomposition. 1. INTRODUCTION By a boundary method, it is usually understood a procedure for solving partial differential equa- tions and/or systems of such equations, in which a subregion or the entire region, is left out of the numerical treatment, by use of available analytical solutions (or, more generally, previously computed solutions). Boundary methods reduce the dimensions involved in the problems, leading to considerable economy of work and constitute a very convenient manner for treating adequately unbounded regions. Generally, the dimensionality of the problem is reduced by one, but even when part of the region is treated by finite elements, the size of the discretized domain is reduced [1,2]. There are two main approaches to formulating boundary methods; one is based on boundary integral equations and the other one, on the use of complete systems of solutions. The author has studied extensively a version of the method based on the use of complete systems of solutions, known as Trefftz method [3–6]. Although Trefftz’s original formulation was linked to a variational principle, this is not required. What is peculiar of Trefftz method, is that solutions of the homoge- nous differential equation — more generally, adjoint differential equation — are used as weighting functions. The method has been used in many fields. For example, applications to Laplace’s equation are given by Mikhlin [7], to the biharmonic equation by Rektorys [8] and to elasticity by Kupradze [9]. Also, many scattered contributions to the method can be found in the literature. Special mention is made here of work by Amerio, Fichera, Kupradze, Picone and Vekua [10–14]. Colton has constructed families of solutions which are complete for parabolic equations [15]. Some years ago, the author started a systematic research of Trefftz method oriented to: clarify the theoretical foundations required for using complete systems of solutions in a reliable manner, and expand the versatility of the method, making it applicable to any problem which is governed by partial differential equations and/or systems of such equations which are linear. For symmetric systems, the results obtained were presented in several reports [5, 6, 16–24] and later integrated in book form [4]. They include: a) a criterium of completeness (introduced in [16] and called Trefftz–Herrera, or TH-completeness); b) approximating procedures and conditions for their convergence [5, 6, 18]; c) formulation of variational principles [19, 20, 24]; and d) development