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

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 -space, . The greater generality of this wea...

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Bibliographic Details
Published in:Journal of computational methods in applied mathematics Vol. 19; no. 3; pp. 557 - 579
Main Authors: Muga, Ignacio, Tyler, Matthew J. W., van der Zee, Kristoffer G.
Format: Journal Article
Language:English
Published: Minsk De Gruyter 01.07.2019
Walter de Gruyter GmbH
Subjects:
35J25
35L02
41A65
46B20
65J05
65N12
65N15
Advection
Advection-Reaction
Approximation
Banach Spaces
Compatible FE Pairs
DDMRes
Discrete Dual Norms
Fortin Condition
Norms
Optimization
Petrov–Galerkin Method
Qualitative analysis
Residual Minimization
ISSN:1609-4840, 1609-9389
Online Access:Get full text
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Summary: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 -space, . 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 , and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection.
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ISSN:1609-4840
1609-9389
DOI:10.1515/cmam-2018-0199