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 $L^q$-type Sobolev spaces, with $1<q<\infty$. We then apply a non-standard, non-linear Petrov–Galerkin 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 $L^2$-setting to the $L^q$-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.

在本文中，我们考虑对流扩散反应方程的数值近似。设计用于此问题的数值方法的主要挑战之一是，在以对流为主的情况下出现的边界层会导致数值逼近中的非物理振荡，通常称为吉布斯现象。本文的思想是将近似问题视为对偶范数中的残差极小${\mathrm{大号}}^q$型Sobolev空间，具有 $\mathrm{1个}<q<\infty$。然后，我们应用非标准，非线性的Petrov-Galerkin离散化，该离散化适用于自反Banach空间，使得该空间本身及其对偶空间都是严格凸的。与不连续的Petrov-Galerkin方法类似，该方法基于最小化对偶范数中的残差。用适当的离散对偶范数代替难对偶范数会导致非线性不精确混合方法。这将在不连续的Petrov-Galerkin方法的背景下开发的Petrov-Galerkin框架推广到更一般的Banach空间。对于对流-扩散-反应方程式，可以得出类似的方法的推广。${\mathrm{大号}}^2$-设置为 ${\mathrm{大号}}^q$-设置。考虑更一般的Banach空间设置的主要优点是，在某些情况下，数值逼近的振荡会随着$q$ 趋于1，正如我们将使用一些简单的数值示例进行演示。