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, Ignacio
AU - Roggendorf, Sarah
AU - Van Der Zee, Kristoffer G.
N1 - Funding Information:
The work by Ignacio Muga was done in the framework of Chilean FONDECYT research project #1160774. Ignacio Muga was also partially supported by the European Union's Horizon 2020, research and innovation program under the Marie Sklodowska-Curie grant agreement No 777778.
Funding Information:
Funding: The work by Ignacio Muga was done in the framework of Chilean FONDECYT research project #1160774. Ignacio Muga was also partially supported by the European Union’s Horizon 2020, research and innovation program under the Marie Sklodowska-Curie grant agreement No 777778.
Publisher Copyright:
© 2019 Walter de Gruyter GmbH, Berlin/Boston 2019.
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 -