{
  "id": "1908.00996",
  "version": "v1",
  "published": "2019-08-02T18:18:16.000Z",
  "updated": "2019-08-02T18:18:16.000Z",
  "title": "Eliminating Gibbs Phenomena: A Non-linear Petrov-Galerkin Method for the Convection-Diffusion-Reaction Equation",
  "authors": [
    "Paul Houston",
    "Sarah Roggendorf",
    "Kristoffer G. van der Zee"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-02T18:18:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30",
      "35J20"
    ],
    "keywords": [
      "convection-diffusion-reaction equation",
      "non-linear petrov-galerkin method",
      "eliminating gibbs phenomena",
      "dual norm",
      "general banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}