The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces

Muga, Ignacio; Tyler, Matthew J. W.; van der Zee, Kristoffer G.

Abstract

© 2019 Walter de Gruyter GmbH, Berlin/Boston 2019. We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue Lp, 1 < p < ∞. The greater generality of this weak setting is natural when dealing with rough data and highly irregular solutions, and when enhanced qualitative features of the approximations are needed. We first present a rigorous analysis of the well-posedness of the underlying continuous weak formulation, under natural assumptions on the advection-reaction coefficients. The main contribution is the study of several discrete subspace pairs guaranteeing the discrete stability of the method and quasi-optimality in L p {L^{p}}, and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection.

Citation

Muga, I., Tyler, M. J., & van der Zee, K. G. (2019). The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces. Computational Methods in Applied Mathematics, 19(3), 557-579. https://doi.org/10.1515/cmam-2018-0199

• Journal Article Type: Article
• Acceptance Date: Jun 13, 2019
• Publication Date: Jul 1, 2019
• Deposit Date: Jul 5, 2019
• Publicly Available Date: Jul 2, 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: 557-579
• DOI: https://doi.org/10.1515/cmam-2018-0199
• Keywords: Applied Mathematics; Numerical Analysis; Computational Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/2273034
• Publisher URL: https://www.degruyter.com/view/j/cmam.2019.19.issue-3/cmam-2018-0199/cmam-2018-0199.xml