Indirect methods of collocation: Trefftz‐Herrera collocation

A nonstandard collocation method (TH‐collocation) is presented, where collocation is used to construct specialized weighting functions instead of the solution itself, as it is usual, so that in this sense it is an indirect method. TH‐collocation is shown to be as accurate as standard collocation, but computationally far more efficient. The present article is the first of a series devoted to explore thoroughly collocation methods. The following classification of collocation methods is introduced: direct‐nonoverlapping; indirect‐nonoverlapping; direct‐overlapping; and indirect‐overlapping. Most of the effort reported in the literature has gone to direct‐nonoverlapping methods. The procedure presented in this article falls into the indirect‐nonoverlapping category and it is based on Trefftz‐Herrera formulation. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 709–738, 1999