Research Repository

# Eliminating Gibbs phenomena: A non-linear Petrov–Galerkin method for the convection–diffusion–reaction equation

## Authors

### Abstract

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.

### Citation

Houston, P., Roggendorf, S., & G. van der Zee, K. (2020). Eliminating Gibbs phenomena: A non-linear Petrov–Galerkin method for the convection–diffusion–reaction equation. Computers and Mathematics with Applications, 80(5), 851-873. https://doi.org/10.1016/j.camwa.2020.03.025

Journal Article Type Article Mar 30, 2020 May 28, 2020 Sep 1, 2020 May 29, 2020 May 29, 2021 Computers & Mathematics with Applications 0898-1221 Elsevier Peer Reviewed 80 5 851-873 https://doi.org/10.1016/j.camwa.2020.03.025 Convection–diffusion; Petrov–Galerkin; Gibbs phenomenon; Finite element methods; Banach spaces https://nottingham-repository.worktribe.com/output/2463138 https://www.sciencedirect.com/science/article/pii/S0898122120301292?via%3Dihub