We introduce and study a new class of projection methods---namely, the velocity-correction methods in standard form and in rotational form---for solving the unsteady incompressible Navier--Stokes equations. We show that the rotational form provides improved error estimates in terms of the H1 -norm for the velocity and of the L2 -norm for the pressure. We also show that the class of fractional-step methods introduced in S. A. Orsag, M. Israeli, and M. Deville, J. Sci. Comput., 1 (1986), pp. 75--111 and K. E. Karniadakis, M. Israeli, and S. A. Orsag, J. Comput. Phys., 97 (1991), pp. 414--443 can be interpreted as the rotational form of our velocity-correction methods. Thus, to the best of our knowledge, our results provide the first rigorous proof of stability and convergence of the methods in those papers. We also emphasize that, contrary to those of the above groups, our formulations are set in the standard L2 setting, and consequently they can be easily implemented by means of any variational approximation techniques, in particular the finite element methods.
%0 Journal Article
%1 2003SINUM..41..112G
%A Guermond, J. L.
%A Shen, Jie
%D 2003
%E 65M12 35Q30, 35J05, 76D05
%I Society for Industrial & Applied Mathematics (SIAM)
%J SIAM Journal on Numerical Analysis
%K 35q30-navier-stokes-equations 65m12-pdes-ivps-stability-and-convergence-of-numerical-methods 76d05-incompressible-navier-stokes-equations
%N 1
%P 112--134
%R 10.1137/s0036142901395400
%T Velocity-Correction Projection Methods for Incompressible Flows
%U https://epubs.siam.org/doi/10.1137/S0036142901395400
%V 41
%X We introduce and study a new class of projection methods---namely, the velocity-correction methods in standard form and in rotational form---for solving the unsteady incompressible Navier--Stokes equations. We show that the rotational form provides improved error estimates in terms of the H1 -norm for the velocity and of the L2 -norm for the pressure. We also show that the class of fractional-step methods introduced in S. A. Orsag, M. Israeli, and M. Deville, J. Sci. Comput., 1 (1986), pp. 75--111 and K. E. Karniadakis, M. Israeli, and S. A. Orsag, J. Comput. Phys., 97 (1991), pp. 414--443 can be interpreted as the rotational form of our velocity-correction methods. Thus, to the best of our knowledge, our results provide the first rigorous proof of stability and convergence of the methods in those papers. We also emphasize that, contrary to those of the above groups, our formulations are set in the standard L2 setting, and consequently they can be easily implemented by means of any variational approximation techniques, in particular the finite element methods.
@article{2003SINUM..41..112G,
abstract = {We introduce and study a new class of projection methods---namely, the velocity-correction methods in standard form and in rotational form---for solving the unsteady incompressible Navier--Stokes equations. We show that the rotational form provides improved error estimates in terms of the H1 -norm for the velocity and of the L2 -norm for the pressure. We also show that the class of fractional-step methods introduced in [S. A. Orsag, M. Israeli, and M. Deville, J. Sci. Comput., 1 (1986), pp. 75--111] and [K. E. Karniadakis, M. Israeli, and S. A. Orsag, J. Comput. Phys., 97 (1991), pp. 414--443] can be interpreted as the rotational form of our velocity-correction methods. Thus, to the best of our knowledge, our results provide the first rigorous proof of stability and convergence of the methods in those papers. We also emphasize that, contrary to those of the above groups, our formulations are set in the standard L2 setting, and consequently they can be easily implemented by means of any variational approximation techniques, in particular the finite element methods.},
added-at = {2019-11-04T00:46:19.000+0100},
author = {Guermond, J. L. and Shen, Jie},
biburl = {https://www.bibsonomy.org/bibtex/270987253176c1d81c2c661eb4bdaca17/gdmcbain},
doi = {10.1137/s0036142901395400},
editor = {{65M12 35Q30, 35J05}, 76D05},
interhash = {4405f005bf15ff97040b1f906b8ccdf3},
intrahash = {70987253176c1d81c2c661eb4bdaca17},
journal = {{SIAM} Journal on Numerical Analysis},
keywords = {35q30-navier-stokes-equations 65m12-pdes-ivps-stability-and-convergence-of-numerical-methods 76d05-incompressible-navier-stokes-equations},
month = jan,
number = 1,
pages = {112--134},
publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})},
timestamp = {2019-11-04T00:46:19.000+0100},
title = {Velocity-Correction Projection Methods for Incompressible Flows},
url = {https://epubs.siam.org/doi/10.1137/S0036142901395400},
volume = 41,
year = 2003
}