New ideas in adjoint methods for PDEs : a saddle-point paradigm for finite element analysis and its role in the DPG methodology

This dissertation presents a novel framework for the construction and analysis of finite element methods with trial and test spaces of unequal dimension. At the heart of this work is a new duality theory suitable for variational formulations with non-symmetric functional settings. The primary application of this theory, in this dissertation, is the development and analysis of discontinuous Petrov–Galerkin (DPG) finite element methods.

This dissertation introduces the DPG* finite element method: the dual to the DPG method. DPG, as a methodology, can be viewed as a practical means to solve overdetermined discretizations of boundary value problems. In a similar way, DPG* delivers a methodology for underdetermined discretizations. Supporting this new finite element method are new results on a priori error estimation and a posteriori error control. Notably, it is demonstrated that the convergence of a DPG* method is controlled, in part, by a Lagrange multiplier variable which plays the role of the solution variable in DPG methods. An important new result on a posteriori error control for DPG methods and comparisons with other related methods are also featured.

The theory developed here is applied to two representative problems coming from linear and nonlinear partial differential equation (PDE) models. To facilitate a thorough mathematical analysis, Poisson's equation is considered. To demonstrate the utility of the approach in less tractable scenarios, the Oldroyd-B fluid model is also considered. Taken together, the combined analysis of these two models effectively demonstrates the utility of the newly developed paradigm.

Extensive computational experiments support the theoretical work presented in this dissertation. In these experiments, h- and hp-adaptive mesh refinement play a central role. For standard solution-oriented adaptive mesh refinement, local error contributions coming from a global a posteriori error estimate are selected to mark individual elements for refinement. For goal-oriented adaptive mesh refinement, local contributions coming from both a primal (DPG) and a dual (or adjoint; DPG*) problem are combined to deliver effective refinement strategies for linear output functionals, also known as quantities of interest.