  In this article we consider the numerical approximation of the
convection-diffusion-reaction equation. One of the main challenges of designing
a numerical method for this problem is that boundary layers occurring in the
convection-dominated case can lead to non-physical oscillations in the
numerical approximation, often referred to as Gibbs phenomena. The idea of this
article is to consider the approximation problem as a residual minimization in
dual norms in Lq-type Sobolev spaces, with 1 < q < $\infty$. We then apply a
non-standard, non-linear PetrovGalerkin discretization, that is applicable to
reflexive Banach spaces such that the space itself and its dual are strictly
convex. Similar to discontinuous Petrov-Galerkin methods, this method is based
on minimizing the residual in a dual norm. Replacing the intractable dual norm
by a suitable discrete dual norm gives rise to a non-linear inexact mixed
method. This generalizes the Petrov-Galerkin framework developed in the context
of discontinuous Petrov-Galerkin methods to more general Banach spaces. For the
convection-diffusion-reaction equation, this yields a generalization of a
similar approach from the L2-setting to the Lq-setting. A key advantage of
considering a more general Banach space setting is that, in certain cases, the
oscillations in the numerical approximation vanish as q tends to 1, as we will
demonstrate using a few simple numerical examples.
