ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis ISBN 978-3-7001-8258-0 Online Edition Research Article
Jessika Camaño,
Cristian Muñoz,
Ricardo Oyarzúa
S. 114 - 130 doi:10.1553/etna_vol48s114 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol48s114
Abstract: In this paper we analyze the numerical approximation of a saddle-point problem posed innon-standard Banach spaces $\mathrm{H}(\mathrm{div}_{p}\,, \Omega)\times L^q(\Omega)$, where$\mathrm{H}(\mathrm{div}_{p}\,, \Omega):= \{{\boldsymbol\tau} \in [L^2(\Omega)]^n \colon \mathrm{div} {\boldsymbol\tau} \in L^p(\Omega)\},$with $p>1$ and $q\in \mathbb{R}$ being the conjugate exponent of $p$ and $\Omega\subseteq \mathbb{R}^n$ ($n\in\{2,3\}$)a bounded domain with Lipschitz boundary $\Gamma$. In particular, we are interestedin deriving the stability properties of the forms involved (inf-sup conditions, boundedness),which are the main ingredients to analyze mixed formulations. In fact, by using these propertieswe prove the well-posedness of the corresponding continuous and discrete saddle-point problems by means of theclassical Babuška-Brezzi theory, where the associated Galerkin scheme is defined by Raviart-Thomaselements of order $k\geq 0$ combined with piecewise polynomials of degree $k$. In addition weprove optimal convergence of the numerical approximation in the associated Lebesgue norms.Next, by employing the theory developed for the saddle-point problem, weanalyze a mixed finite element method for a convection-diffusion problem, providing well-posedness of the continuousand discrete problems and optimal convergence under a smallness assumption on the convective vector field.Finally, we corroborate the theoretical results with suitable numerical results in two and three dimensions. Keywords: mixed finite element method, Raviart-Thomas, Lebesgue spaces, Lp data, convection-diffusion Published Online: 2018/04/19 09:51:16 Document Date: 2018/04/19 07:57:00 Object Identifier: 0xc1aa5576 0x003892e4 Rights: . Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613. …
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |