Spatial Error Estimates for a Finite Element Viscosity-Splitting Scheme for the Navier-Stokes Equations
Year: 2013
Author: F. Guillen-Gonzalez, M. V. Redondo-Neble
International Journal of Numerical Analysis and Modeling, Vol. 10 (2013), Iss. 4 : pp. 826–844
Abstract
In this paper, we obtain optimal first order error estimates for a fully discrete fractional-step scheme applied to the Navier-Stokes equations. This scheme uses decomposition
of the viscosity in time and finite elements (FE) in space.
In [15], optimal first order error estimates (for velocity and pressure) for the corresponding time-discrete scheme were obtained, using in particular $H^2 \times H^1$ estimates for the approximations of
the velocity and pressure. Now, we use this time-discrete scheme as an auxiliary problem to study
a fully discrete finite element scheme, obtaining optimal first order approximation for velocity and
pressure with respect to the max-norm in time and the $H^1 \times L^2$-norm in space.
The proof of these error estimates are based on three main points: a) provide some new estimates
for the time-discrete scheme (not proved in [15]) which must be now used, b) give a discrete version
of the $H^2 \times H^1$ estimates in FE spaces, using stability in the $W^{1,6} \times L^6$-norm of the FE Stokes
projector, and c) the use of a weight function vanishing at initial time will let to hold the error
estimates without imposing global compatibility for the exact solution.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2013-IJNAM-598
International Journal of Numerical Analysis and Modeling, Vol. 10 (2013), Iss. 4 : pp. 826–844
Published online: 2013-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 19
Keywords: Navier-Stokes Equations splitting in time schemes fully discrete schemes error estimates mixed formulation stable finite elements.