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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Fr├ęd├ęric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold


IMPACT FACTOR 2018: 3.107

CiteScore 2018: 2.43

SCImago Journal Rank (SJR) 2018: 1.252
Source Normalized Impact per Paper (SNIP) 2018: 1.618

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1569-3953
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Volume 25, Issue 4

Issues

On three steps two-grid finite element methods for the 2D-transient Navier-Stokes equations

Saumya Bajpai / Amiya K. Pani
  • Corresponding author
  • Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India
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Published Online: 2018-02-22 | DOI: https://doi.org/10.1515/jnma-2016-1055

Abstract

In this paper, an error analysis of a three steps two level Galerkin finite element method for the two dimensional transient NavierÔÇôStokes equations is discussed. First of all, the problem is discretized in spatial direction by employing finite element method on a coarse mesh ­Łĺ»H with mesh size H. Then, in step two, the nonlinear system is linearized around the coarse grid solution, say, uH, which is similar to NewtonÔÇÖs type iteration and the resulting linear system is solved on a finer mesh ­Łĺ»h with mesh size h. In step three, a correction is obtained through solving a linear problem on the finer mesh and an updated final solution is derived. Optimal error estimates in LÔł×(L2)-norm, when h = ­Łĺ¬ (H2Ôłĺ╬┤) and in LÔł×(H1)-norm, when h = ­Łĺ¬(H4Ôłĺ╬┤) for the velocity and in LÔł×(L2)-norm, when h = ­Łĺ¬(H4Ôłĺ╬┤) for the pressure are established for arbitrarily small ╬┤ > 0. Further, under uniqueness assumption, these estimates are proved to be valid uniformly in time. Then, based on backward Euler method, a completely discrete scheme is analyzed and a priori error estimates are derived. Results obtained in this paper are sharper than those derived earlier by two-grid methods. Finally, the paper is concluded with some numerical experiments.

Keywords: two-grid method; 2D Navier-Stokes system; semidiscrete scheme; backward Euler method; optimal error estimates; order of convergence; uniform-in-time estimates; uniqueness assumption; numerical experiments

MSC 2010: 65M60; 65M12; 65M15; 35D05; 35D10

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About the article

Received: 2016-06-10

Revised: 2017-01-29

Accepted: 2017-01-29

Published Online: 2018-02-22

Published in Print: 2017-12-20


Citation Information: Journal of Numerical Mathematics, Volume 25, Issue 4, Pages 199ÔÇô228, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI:┬áhttps://doi.org/10.1515/jnma-2016-1055.

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