TY - JOUR

T1 - The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces

T2 - A Direct Proof of the Inf-Sup Condition and Stability of Galerkin's Method

AU - Houston, Paul

AU - MUGA URQUIZA, IGNACIO PATRICIO PEDRO

AU - Roggendorf, Sarah

AU - Van Der Zee, Kristoffer G.

PY - 2019/7/1

Y1 - 2019/7/1

N2 - 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.

AB - 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.

KW - Banach Spaces

KW - Convection-Diffusion Equation

KW - Elliptic Regularity

KW - FEM

KW - Galerkin Methods

KW - Inf-Sup Condition

KW - Well-Posedness

UR - http://www.scopus.com/inward/record.url?scp=85068418087&partnerID=8YFLogxK

U2 - 10.1515/cmam-2018-0198

DO - 10.1515/cmam-2018-0198

M3 - Article

AN - SCOPUS:85068418087

VL - 19

SP - 503

EP - 522

JO - Computational Methods in Applied Mathematics

JF - Computational Methods in Applied Mathematics

SN - 1609-4840

IS - 3

ER -