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Error estimates for semi-discrete gauge methods for the Navier-Stokes equations

Authors: Ricardo H. Nochetto and Jae-Hong Pyo
Journal: Math. Comp. 74 (2005), 521-542
MSC (2000): Primary 65M12, 65M15, 76D05
Published electronically: July 20, 2004
MathSciNet review: 2114636
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Abstract: The gauge formulation of the Navier-Stokes equations for incompressible fluids is a new projection method. It splits the velocity $\mathbf{u}=\mathbf{a}+\nabla\phi$ in terms of auxiliary (nonphysical) variables $\mathbf{a}$ and $\phi$ and replaces the momentum equation by a heat-like equation for $\mathbf{a}$ and the incompressibility constraint by a diffusion equation for $\phi$. This paper studies two time-discrete algorithms based on this splitting and the backward Euler method for $\mathbf{a}$ with explicit boundary conditions and shows their stability and rates of convergence for both velocity and pressure. The analyses are variational and hinge on realistic regularity requirements on the exact solution and data. Both Neumann and Dirichlet boundary conditions are, in principle, admissible for $\phi$ but a compatibility restriction for the latter is uncovered which limits its applicability.

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Additional Information

Ricardo H. Nochetto
Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742

Jae-Hong Pyo
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Keywords: Projection method, Gauge method, Navier-Stokes equation, incompressible fluids
Received by editor(s): March 21, 2003
Received by editor(s) in revised form: October 20, 2003
Published electronically: July 20, 2004
Additional Notes: The first author was partially supported by NSF Grants DMS-9971450 and DMS-0204670.
The second author was partially supported by NSF Grant DMS-9971450
Article copyright: © Copyright 2004 American Mathematical Society