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