Korean J. Math.  Vol 21, No 2 (2013)  pp.125-150
DOI: https://doi.org/10.11568/kjm.2013.21.2.125

Error estimates for the Fully discrete stabilized Gauge-Uzawa Method/Part I: The Navier-Stokes equations

Jae-Hong Pyo

Abstract


The stabilized Gauge-Uzawa method (SGUM), which is a second order projection type algorithm   to solve the time-dependent Navier-Stokes equations, has been newly constructed in   2013 Pyo's paper. The accuracy of SGUM has been proved only for time   discrete scheme in the same paper,   but it is crucial to study for fully discrete scheme,   because the numerical errors depend on discretizations for both space   and time,   and because discrete spaces between velocity and pressure can not be   chosen arbitrary.   In this paper, we find out properties of the fully discrete SGUM and   estimate its errors and stability to solve the evolution Navier-Stokes equations.   The main difficulty in this estimation arises from losing some cancellation laws   due to failing divergence free condition of the discrete velocity function.   This result will be extended to Boussinesq equations in the continuous research (part II)   and is essential in the study of part II.

Subject classification

65M12, 65M15, 76D05

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ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr