English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert spaceH  0  1   ⁢   (  Ω  ){H_{0}^{1}(\Omega)}   , the Banach Sobolev spaceW  0   1  ,  q    ⁢   (  Ω  ){W^{1,q}_{0}(\Omega)}   ,1  &lt;  q  &lt;  ∞{1&lt;q&lt;{\infty}}   , is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in theW  0   1  ,  q    ⁢   (  Ω  ){W^{1,q}_{0}(\Omega)}   -W  0   1  ,   q  ′⁢   (  Ω  ){W_{0}^{1,q^{\prime}}(\Omega)}    functional setting,1  q   +   1   q  ′=  1{\frac{1}{q}+\frac{1}{q^{\prime}}=1}   . 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 ofW   1  ,   q  ′{W^{1,q^{\prime}}}   -stability of theH  0  1{H_{0}^{1}}   -projector.

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