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Some discrete inequalities for central-difference type operators


Authors: Hiroki Kojima, Takayasu Matsuo and Daisuke Furihata
Journal: Math. Comp. 86 (2017), 1719-1739
MSC (2010): Primary 65M06, 65M12
DOI: https://doi.org/10.1090/mcom/3154
Published electronically: September 15, 2016
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Abstract: Discrete versions of basic inequalities in functional analysis such as the Sobolev inequality play a key role in theoretical analysis of finite difference schemes. They have been shown for some simple difference operators, but are still left open for general operators, even including the standard central difference operators. In this paper, we propose a systematic approach for deriving such inequalities for a certain class of central-difference type operators. We illustrate the results by giving a generic a priori estimate for certain conservative schemes for the nonlinear Schrödinger and Cahn-Hilliard equations.


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Additional Information

Hiroki Kojima
Affiliation: Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, 113-8656, Japan
Email: hiroki_kojima@mist.i.u-tokyo.ac.jp, h.k.psi.mot@gmail.com

Takayasu Matsuo
Affiliation: Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, 113-8656, Japan
Email: matsuo@mist.i.u-tokyo.ac.jp

Daisuke Furihata
Affiliation: Cybermedia Center, Osaka University, 1-32 Machikaneyama-cho, Toyonaka City, Osaka 560-0043, Japan
Email: furihata@cmc.osaka-u.ac.jp

DOI: https://doi.org/10.1090/mcom/3154
Received by editor(s): February 16, 2015
Received by editor(s) in revised form: November 28, 2015
Published electronically: September 15, 2016
Additional Notes: This work was supported by JSPS KAKENHI Grant Number 23560063 and 25287030. This work was also supported by CREST, JST
Article copyright: © Copyright 2016 American Mathematical Society