  We propose and analyse a minimal-residual method in discrete dual norms for
approximating the solution of the advection-reaction equation in a weak
Banach-space setting. The weak formulation allows for the direct approximation
of solutions in the Lebesgue $L^p$-space, $1<p<\infty$. The greater generality
of this weak setting is natural when dealing with rough data and highly
irregular solutions, and when enhanced qualitative features of the
approximations are needed.
  We first present a rigorous analysis of the well-posedness of the underlying
continuous weak formulation, under natural assumptions on the
advection-reaction coefficients. The main contribution is the study of several
discrete subspace pairs guaranteeing the discrete stability of the method and
quasi-optimality in $L^p$, and providing numerical illustrations of these
findings, including the elimination of Gibbs phenomena, computation of optimal
test spaces, and application to 2-D advection.
