# Convergence of approximate solutions to the transport equation

We are interested in solving linear transport problems by proving convergence of sequences of approximate solutions. We split the problem into two parts: geometry and transport. In the geometry part, we first construct the flow associated to the velocity field, which we assume to be Lipschitz continuous and bounded. Classical results from the theory of ordinary differential equations then guarantee the existence, uniqueness and stability of flow solutions. For the transport part, we consider the problem along characteristic curves, which leads to a simplified transport problem where the velocity field becomes constant. To investigate the convergence of (approximate) solutions for this problem, we look at a suitable variational formulation and construct stable Petrov-Galerkin methods. To this end, we study the \emph{trial-to-test} map that allows us to prove uniform and almost optimal $\inf$-$\sup$-conditions. Existence, uniqueness, and error estimates then follow from classical results based on the Banach-Ne\u{c}as the oremand Cea's lemma.