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

Houston, Paul; Muga, Ignacio; Roggendorf, Sarah; van der Zee, Kristoffer

Abstract

While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H10(Ω), the Banach Sobolev space W1,q0(Ω), 1 less than ∞ , is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the W1,q0(Ω)-W1,q′0(Ω) functional setting, 1q+1q′=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 of W1,q′-stability of the H10-projector.

Citation

Houston, P., Muga, I., Roggendorf, S., & van der Zee, K. (2019). The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method. Computational Methods in Applied Mathematics, 19(3), 503-522. https://doi.org/10.1515/cmam-2018-0198

• Journal Article Type: Article
• Acceptance Date: Apr 30, 2019
• Online Publication Date: Jun 27, 2019
• Publication Date: Jun 27, 2019
• Deposit Date: May 19, 2019
• Publicly Available Date: Jun 28, 2020
• Journal: Computational Methods in Applied Mathematics
• Print ISSN: 1609-4840
• Electronic ISSN: 1609-9389
• Publisher: De Gruyter
• Peer Reviewed: Peer Reviewed
• Volume: 19
• Issue: 3
• Pages: 503-522
• DOI: https://doi.org/10.1515/cmam-2018-0198
• Keywords: Applied Mathematics; Numerical Analysis; Computational Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/2064818
• Publisher URL: https://www.degruyter.com/view/j/cmam.2019.19.issue-3/cmam-2018-0198/cmam-2018-0198.xml
• Contract Date: May 19, 2019