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Trivariate spline approximations of 3D Navier-Stokes equations

Authors: Gerard Awanou and Ming-Jun Lai
Journal: Math. Comp. 74 (2005), 585-601
MSC (2000): Primary 65D07, 65D15, 35Q30, 76D05
Published electronically: September 2, 2004
MathSciNet review: 2114639
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Abstract: We present numerical approximations of the 3D steady state Navier-Stokes equations in velocity-pressure formulation using trivariate splines of arbitrary degree $d$ and arbitrary smoothness $r$ with $r<d$. Using functional arguments, we derive the discrete Navier-Stokes equations in terms of $B$-coefficients of trivariate splines over a tetrahedral partition of any given polygonal domain. Smoothness conditions, boundary conditions and the divergence-free condition are enforced through Lagrange multipliers. The pressure is computed by solving a Poisson equation with Neumann boundary conditions. We have implemented this approach in MATLAB and present numerical evidence of the convergence rate as well as experiments on the lid driven cavity flow problem.

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Additional Information

Gerard Awanou
Affiliation: The Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455-0436

Ming-Jun Lai
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Keywords: Multivariate splines, numerical solution, Navier-Stokes equations, Lagrange multiplier method, augmented Lagrangian algorithm
Received by editor(s): July 26, 2002
Received by editor(s) in revised form: April 19, 2003
Published electronically: September 2, 2004
Additional Notes: The second author was supported by the National Science Foundation under grant #EAR-0327577 and the Army Research Office under grant DAAD19-03-1-0203. The author wishes to thank the funding agencies for their generosity.
Article copyright: © Copyright 2004 American Mathematical Society