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

Houston, Paul; Roggendorf, Sarah; van der Zee, Kristoffer G.

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<∞. 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 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., & van der Zee, K. G. (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
• Acceptance Date: Mar 30, 2020
• Online Publication Date: May 28, 2020
• Publication Date: Sep 1, 2020
• Deposit Date: May 29, 2020
• Publicly Available Date: May 29, 2021
• Journal: Computers and Mathematics with Applications
• Print ISSN: 0898-1221
• Electronic ISSN: 1873-7668
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 80
• Issue: 5
• Pages: 851-873
• DOI: https://doi.org/10.1016/j.camwa.2020.03.025
• Keywords: Convection–diffusion; Petrov–Galerkin; Gibbs phenomenon; Finite element methods; Banach spaces
• Public URL: https://nottingham-repository.worktribe.com/output/2463138
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0898122120301292?via%3Dihub