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Discretization of linear problems in banach spaces: Residual minimization, nonlinear petrov-galerkin, and monotone mixed methods

Muga, Ignacio; Van Der Zee, Kristoffer G.

Discretization of linear problems in banach spaces: Residual minimization, nonlinear petrov-galerkin, and monotone mixed methods Thumbnail


Authors

Ignacio Muga

KRISTOFFER VAN DER ZEE KG.VANDERZEE@NOTTINGHAM.AC.UK
Professor of Numerical Analysis &computational Applied Mathematics



Abstract

This work presents a comprehensive discretization theory for abstract linear operator equations in Banach spaces. The fundamental starting point of the theory is the idea of residual minimization in dual norms and its inexact version using discrete dual norms. It is shown that this development, in the case of strictly convex reflexive Banach spaces with strictly convex dual, gives rise to a class of nonlinear Petrov-Galerkin methods and, equivalently, abstract mixed methods with monotone nonlinearity. Under the Fortin condition, we prove discrete stability and quasioptimal convergence of the abstract inexact method, with constants depending on the geometry of the underlying Banach spaces. The theory generalizes and extends the classical Petrov-Galerkin method as well as existing residual-minimization approaches, such as the discontinuous Petrov- Galerkin method.

Citation

Muga, I., & Van Der Zee, K. G. (2020). Discretization of linear problems in banach spaces: Residual minimization, nonlinear petrov-galerkin, and monotone mixed methods. SIAM Journal on Numerical Analysis, 58(6), 3406-3426. https://doi.org/10.1137/20M1324338

Journal Article Type Article
Acceptance Date Sep 14, 2020
Online Publication Date Nov 24, 2020
Publication Date Nov 24, 2020
Deposit Date Sep 18, 2020
Publicly Available Date Mar 29, 2024
Journal SIAM Journal on Numerical Analysis
Print ISSN 0036-1429
Electronic ISSN 1095-7170
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 58
Issue 6
Pages 3406-3426
DOI https://doi.org/10.1137/20M1324338
Keywords Operators in Banach spaces, Residual minimization, Petrov–Galerkin discretization, Error analysis, Quasi-optimality, Duality mapping, Best approximation, Geometric constants
Public URL https://nottingham-repository.worktribe.com/output/2463148
Publisher URL https://epubs.siam.org/doi/10.1137/20M1324338

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