In numerical analysis, finite element methods are a method of approximating solutions to differential equations on a domain. In such methods, the solution function is approximated by partitioning the domain into a mesh of elements, and testing candidate functions in a discrete trial space on that mesh against a discrete space of test functions. We explore certain classes of finite element methods called discontinuous Petrov-Galerkin (DPG) finite element methods, where the test space functions are allowed to be discontinuous across elements, and test spaces are selected specifically to optimize stability.

Because we are concerned with the accuracy of our approximation, we place focus on how the error behaves in DPG methods. We explore how DPG methods in semi-linear problems, as well as how DPG problems can interact with adaptive methods, a different framework for finite element methods. In addition, we establish some results about the error of DPG approximations, particularly the error using the subspace dual norms that arise from the construction of the test spaces.