A Lagrange-Galerkin scheme with a locally linearized velocity for the Navier--Stokes equations
We present a Lagrange--Galerkin scheme free from numerical quadrature for the Navier--Stokes equations. Our idea is to use a locally linearized velocity and the backward Euler method in finding the position of fluid particle at the previous time step. Since the scheme can be implemented exactly as i...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
25.05.2015
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Subjects | |
Online Access | Get full text |
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Summary: | We present a Lagrange--Galerkin scheme free from numerical quadrature for the
Navier--Stokes equations. Our idea is to use a locally linearized velocity and
the backward Euler method in finding the position of fluid particle at the
previous time step. Since the scheme can be implemented exactly as it is, the
theoretical stability and convergence results are assured. While the
conventional Lagrange--Galerkin schemes may encounter the instability caused by
numerical quadrature errors, the present scheme is genuinely stable. For the
$\pk 2/\pk 1$- and $\mini$-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 the
genuine stability of the scheme. |
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DOI: | 10.48550/arxiv.1505.06681 |