Houston, Paul; Roggendorf, Sarah; van der Zee, Kristoffer G. (2020). Eliminating Gibbs phenomena: A nonlinear PetrovGalerkin method for the convectiondiffusionreaction equation. Computers and mathematics with applications, 80(5), pp. 851873. Elsevier 10.1016/j.camwa.2020.03.025
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In this article we consider the numerical approximation of the convectiondiffusionreaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convectiondominated case can lead to nonphysical 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^qtype Sobolev spaces, with 1 < q < ∞. We then apply a nonstandard, nonlinear 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 nonlinear 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^2setting to the L^qsetting. 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.
Item Type: 
Journal Article (Original Article) 

Division/Institute: 
08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics 
UniBE Contributor: 
Roggendorf, Sarah 
Subjects: 
500 Science > 510 Mathematics 
ISSN: 
08981221 
Publisher: 
Elsevier 
Language: 
English 
Submitter: 
Sebastiano Don 
Date Deposited: 
10 Feb 2021 17:33 
Last Modified: 
12 Feb 2021 06:41 
Publisher DOI: 
10.1016/j.camwa.2020.03.025 
ArXiv ID: 
1908.00996 
Uncontrolled Keywords: 
Convection–diffusion, Petrov–Galerkin, Gibbs phenomenon, Finite element methods, Banach spaces 
BORIS DOI: 
10.48350/151272 
URI: 
https://boris.unibe.ch/id/eprint/151272 