Five Lectures on Dpg Methods

These lectures present a relatively recent introduction into the class of discontinuos Galerkin (DG) methods, named Discontinuous Petrov-Galerkin (DPG) methods. DPG methods, in which DG spaces form a critical ingredient, can be thought of as least-square methods in nonstandard norms, or as Petrov-Galerkin methods with special test spaces, or as a nonstandard mixed method. We will pursue all these points of view in this lecture. By way of preliminaries, let us recall two results from the Babuška-Brezzi theory. Throughout, bp ̈, ̈q : X ˆ Y Ñ C is a continuous sesquilinear form where X and Y are (generally not equal) normed linear spaces over the complex field C, Y ̊ denotes the space of continuous conjugatelinear functionals on Y , and ` P Y ̊.