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 pape
r is devoted to revise the present state of these matters. Starting fr
om the basic concepts of the algebraic theory, Green-Herrera formulas
are presented and Localized Adjoint Method (LAM) derived. Then the cla
ssical Trefftz method is shown to be a particular case of LAM. This le
ads to a natural generalization of Trefftz method and a special class
of domain decomposition methods: Trefftz-Herrera domain decomposition.