The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin's Method

Paul Houston, IGNACIO PATRICIO PEDRO MUGA URQUIZA, Sarah Roggendorf, Kristoffer G. Van Der Zee

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Abstract

While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space {equation presented}, the Banach Sobolev space{equation presented}, is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the {equation presented}. The theory is based on directly establishing the inf-sup conditions. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplacian. An elementary consequence of the well-posedness theory is the stability and convergence of Galerkin's method in this setting, for a diffusion-dominated case and under the assumption of {equation presented}-projector.

Original languageEnglish
Pages (from-to)503-522
Number of pages20
JournalComputational Methods in Applied Mathematics
Volume19
Issue number3
DOIs
StatePublished - 1 Jul 2019

Keywords

  • Banach Spaces
  • Convection-Diffusion Equation
  • Elliptic Regularity
  • FEM
  • Galerkin Methods
  • Inf-Sup Condition
  • Well-Posedness

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