A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

We present a Petrov-Gelerkin (PG) method for class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm on test space as induced by trial norm, i.e., optimal so that inf-sup condition can be satisfied uniformly independent problem. show well-posedness problems under with general assumptions and convection kernels. Second, following framework Cohen et al.~(2012), embed original problem into larger mixed to choose an enriched stabilization numerical algorithm. In experiments, use approximate which efficiently implemented 1d, study its performance against energy space. conduct convergence studies using uniform $h$- $p$-refinements, adaptive $h$-refinements both smooth manufactured solutions sharp gradient transition layer. addition, confirm PG is asymptotically compatible.