Analysis of a sequential regularization method for the unsteady Navier-Stokes equations

Authors:
Xiliang Lu, Ping Lin and Jian-Guo Liu

Journal:
Math. Comp. **77** (2008), 1467-1494

MSC (2000):
Primary 65M12; Secondary 76D05

DOI:
https://doi.org/10.1090/S0025-5718-08-02087-5

Published electronically:
February 1, 2008

MathSciNet review:
2398777

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Abstract: The incompressibility constraint makes Navier-Stokes equations difficult. A reformulation to a better posed problem is needed before solving it numerically. The sequential regularization method (SRM) is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods. In the paper, we study the existence and uniqueness for the solution of the SRM and provide a simple proof of the convergence of the solution of the SRM to the solution of the Navier-Stokes equations. We also give error estimates for the time discretized SRM formulation.

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Additional Information

**Xiliang Lu**

Affiliation:
Department of Mathematics, National University of Singapore, Science Drive 2, Singapore 117543, Singapore

Address at time of publication:
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Strasse 69, A-4040 Linz, Austria

Email:
lu_xiliang@hotmail.com

**Ping Lin**

Affiliation:
Department of Mathematics, National University of Singapore, Science Drive 2, Singapore 117543, Singapore

Email:
matlinp@nus.edu.sg

**Jian-Guo Liu**

Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742-4015

Email:
jliu@math.umd.edu

DOI:
https://doi.org/10.1090/S0025-5718-08-02087-5

Keywords:
Navier-Stokes equations,
iterative penalty method,
implicit parabolic PDE,
error estimates,
constrained dynamical system,
stabilization method

Received by editor(s):
July 25, 2006

Received by editor(s) in revised form:
June 4, 2007

Published electronically:
February 1, 2008

Additional Notes:
The research was supported by several grants at the Department of Mathematics, National University of Singapore

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.