In this talk, we propose a general framework for variational formulations of first order transport equations in bounded domains and investigate well-posedness and bounded invertibility of the corresponding linear operators. In particular, the variational formulation should allow us to employ directional representation system like shearlets in order to deal with anisotropic features.
A framework for stable discretizations of boundary value problems for these operators is outlined and the proposed stability concept is based on suitable perturbations of certain ideal test spaces. We develop a general strategy for generating such perturbed test spaces based on a certain projection approach. We discuss two consequences, namely stable frame representations for such operator equations as well as an adaptive solution scheme with guaranteed convergence. The results are illustrated by first experiments.