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An exactly computable Lagrange-Galerkin scheme for the Navier-Stokes equations and its error estimates


Authors: Masahisa Tabata and Shinya Uchiumi
Journal: Math. Comp. 87 (2018), 39-67
MSC (2010): Primary 65M12, 65M25, 65M60, 76D05, 76M10
DOI: https://doi.org/10.1090/mcom/3222
Published electronically: May 11, 2017
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Abstract: We present a Lagrange-Galerkin scheme, which is computable exactly, for the Navier-Stokes equations and show its error estimates. In the Lagrange-Galerkin method we have to deal with the integration of composite functions, where it is difficult to get the exact value. In real computations, numerical quadrature is usually applied to the integration to obtain approximate values, that is, the scheme is not computable exactly. It is known that the error caused from the approximation may destroy the stability result that is proved under the exact integration. Here we introduce a locally linearized velocity and the backward Euler method in solving ordinary differential equations in the position of the fluid particle. Then, the scheme becomes computable exactly, and we show the stability and convergence for this scheme. For the $ \mathrm {P}_{2}/\mathrm {P}_{1}$- and $ \mathrm {P_{1}+}/\mathrm {P}_{1}$-finite elements optimal error estimates are proved in $ \ell ^\infty (H^1)\times \ell ^2(L^2)$ norm for the velocity and pressure. We present some numerical results, which reflect these estimates and also show robust stability for high Reynolds numbers in the cavity flow problem.


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Additional Information

Masahisa Tabata
Affiliation: Department of Mathematics, Waseda University, 3-4-1, Ohkubo, Shinjuku, Tokyo 169-8555, Japan
Email: tabata@waseda.jp

Shinya Uchiumi
Affiliation: Research Fellow of Japan Society for the Promotion of Science and Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1, Ohkubo, Shinjuku, Tokyo 169-8555, Japan
Email: su48@fuji.waseda.jp

DOI: https://doi.org/10.1090/mcom/3222
Keywords: Lagrange--Galerkin scheme, finite element method, Navier--Stokes equations, exact computation.
Received by editor(s): September 4, 2015
Received by editor(s) in revised form: August 2, 2016
Published electronically: May 11, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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