I'm taking a class right now (graduate level), an Introduction to Nodal Discontinuous Galerkin methods. I've got a little numerical analysis background, meaning I've covered a lot of topics over the last year but never got really in depth in any single topic. Anyway, the book I'm reading, develops these NDG methods by introducing test functions which are new to me. The book requires that the residual (does residual mean error?) of our approximated solutions (to whatever pde) is orthogonal to these test functions? Why? The only reason I can glean, is that test functions are just cleverly chosen functions, and the orthogonality requirement just makes the coefficient solving easier? Is that the case or is there something deeper here? As a note, the authors of my book chose the test functions to be in the same space as the proposed approximated polynomial solution, both locally and globally.