finite element methods are numerical methods that approximate solutions to pdes using functions on a mesh representing the problem domain. discontinuous-petrov galerkin methods are a class of finite element methods that are aimed at achieving stability of the petrov-galerkin finite element approximation through a careful selection of the associated trial and test spaces. in this talk, i will present dpg theorems as they apply to linear problems, and then approaches for those theorems in the case of non-linear problems, as well as suggest further approaches to non-linear problems.