Discontinuous Petrov-Galerkin methods with optimal test spaces for convection dominated convection-diffusion equations

In this thesis, Discontinuous Petrov-Galerkin (DPG) finite element methods are developed for convection-diffusion equations. In particular, this thesis focuses on the use of optimal test spaces.
A convection-diffusion equation is a singularly perturbed problem. That is, the nature of the problem changes when the diffusion term vanishes, which makes it challenging to solve numerically for small diffusion values, i.e. when convection dominates. Standard finite element methods give very unsatisfactory results, producing approximations that exhibit spurious oscillations and other nonphysical behavior.
Recently, a class of finite element methods has been developed, in which optimal test spaces are used. These spaces guarantee that one gets the best approximation from the trial space in which the solution is sought. The methods are examples of least-squares methods, with the special property that one can choose the norm in which the residual is minimized. This freedom of choice allows us to control the norm in which the best approximation is obtained.
The new approach in this thesis is that the variational formulation associated with the convection-diffusion problem also gives a well-posed variational formulation of the limit convection problem if the diffusion term vanishes. This is necessary in order to retain stability, and to make sure that the computational cost does not grow, when the diffusion term decreases.
Special attention is paid to the transport problem which, besides being the limit problem for vanishing diffusion, also has other applications. A new method is introduced that outperforms existing methods in convergence rates, but also in reducing the smearing of discontinuities of solutions.
The theory developed in this thesis is illustrated by various numerical results.