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
Mohadese Ramezani,
Reza Mokhtari,
Gundolf Haase
S. 568 - 584 doi:10.1553/etna_vol55s568 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol55s568
Abstract: A time-fractional diffusion equation with the Caputo fractional derivative of order α∈(0,1) is considered on a bounded polygonal domain. Some numerical methods are presented based on the finite element method (FEM) in space on a quasi-uniform mesh and L-type discretizations (i.e., L1, L1-2, and L1-2-3 formulas) to approximate the Caputo derivative. Stability and convergence of the L1-2-3 FEM as well as L1-2 FEM are proved rigorously. The lack of positivity of the coefficients of these formulas is the main difficulty in the analysis of the proposed methods. This has hampered the analysis of methods using finite elements mixed with L1-2 and L1-2-3 discretizations. Our proofs are based on the concept of a special kind of discrete Grönwall's inequality and the energy method. Numerical examples confirm the theoretical analysis. Keywords: subdiffusion equation, finite element method, Caputo derivative, L1 formula, L1-2 formula, L1-2-3 formula, Grönwall's inequality, stability analysis, convergence analysis Published Online: 2022/06/10 07:54:42 Object Identifier: 0xc1aa5576 0x003d79ab 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 |