The convection-diffusion Petrov-Galerkin story

The term 'Petrov-Galerkin method' is probably due to Ron Mitchell and his collaborators, and he was certainly very active in studying the application of finite-element methods to second-order partial differential equations 'with significant first derivatives'. Our aim in the present paper is to trace links between such early methods and the more recent discontinuous Galerkin methods-not only the methods but also their analysis. Also as Mitchell, like the author, was initially steeped in finite-difference methods, we shall sometimes use their manipulation in our analysis.