Unified theory of Trefftz methods and numerical implications

In the 1 International Workshop, devoted to Trefftz Method, the author presented an indirect approach to Trefftz Method (Trefftz-Herrera Method), while in the Second one some of the basic ideas of how to integrate different approaches to Trefftz method were introduced. The present Plenary Lecture, corresponding to the 3 International Workshop of this series, is devoted to show that Trefftz Method, when formulated in an suitable framework, is a very broad concept capable of incorporating and unifying many numerical methods for partial differential equations. In this manner, the unified theory of Trefftz Method that was announced in the second publication of this series, has been developed. It includes Direct Trefftz Methods (TrefftzJirousek) and Indirect Trefftz Methods (Trefftz-Herrera). At present, the unified theory is fully developed and an overview is given here, as well as a brief description of its numerical implications.