Volume 10, Issue 4
Spatial Error Estimates for a Finite Element Viscosity-Splitting Scheme for the Navier-Stokes Equations

Int. J. Numer. Anal. Mod., 10 (2013), pp. 826-844.

Published online: 2013-10

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• 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.

• Keywords

Navier-Stokes Equations, splitting in time schemes, fully discrete schemes, error estimates, mixed formulation, stable finite elements.

35Q30, 65N15, 65N30, 76D05

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@Article{IJNAM-10-826, author = {Guillen-Gonzalez , F. and Redondo-Neble , M. V.}, title = {Spatial Error Estimates for a Finite Element Viscosity-Splitting Scheme for the Navier-Stokes Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {4}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/598.html} }
TY - JOUR T1 - Spatial Error Estimates for a Finite Element Viscosity-Splitting Scheme for the Navier-Stokes Equations AU - Guillen-Gonzalez , F. AU - Redondo-Neble , M. V. JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 826 EP - 844 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/598.html KW - Navier-Stokes Equations, splitting in time schemes, fully discrete schemes, error estimates, mixed formulation, stable finite elements. AB -

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.

F. Guillen-Gonzalez & M. V. Redondo-Neble. (1970). Spatial Error Estimates for a Finite Element Viscosity-Splitting Scheme for the Navier-Stokes Equations. International Journal of Numerical Analysis and Modeling. 10 (4). 826-844. doi:
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