Trivariate spline approximations of 3D Navier-Stokes equations

Awanou, Gerard; Lai, Ming-Jun

doi:10.1090/S0025-5718-04-01715-6

Trivariate spline approximations of 3D Navier-Stokes equations
HTML articles powered by AMS MathViewer

by Gerard Awanou and Ming-Jun Lai;

Math. Comp. 74 (2005), 585-601

DOI: https://doi.org/10.1090/S0025-5718-04-01715-6

Published electronically: September 2, 2004

PDF | Request permission

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.

References

G. Awanou, Energy Methods in 3D Spline Approximations, Ph.D. Dissertation, University of Georgia, Athens, Georgia, 2003.
G. Awanou and M. J. Lai, On convergence rate of the augmented Lagrangian algorithm for non-symmetric saddle point problems, to appear, J. Appl. Numer. Math.
G. Awanou, M. J. Lai and P. Wenston, The multivariate spline method for numerical solutions of partial differential equations and scattered data fitting, submitted for publication (2002).
Ivo Babuška, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1972/73), 179–192. MR 359352, DOI 10.1007/BF01436561
Carl de Boor, $B$-form basics, Geometric modeling, SIAM, Philadelphia, PA, 1987, pp. 131–148. MR 936450
Charles K. Chui and Ming Jun Lai, Multivariate vertex splines and finite elements, J. Approx. Theory 60 (1990), no. 3, 245–343. MR 1042653, DOI 10.1016/0021-9045(90)90063-V
K. C. Chung and T. H. Yao, On lattices admitting unique Lagrange interpolations, SIAM J. Numer. Anal. 14 (1977), no. 4, 735–743. MR 445158, DOI 10.1137/0714050
Gerald Farin, Triangular Bernstein-Bézier patches, Comput. Aided Geom. Design 3 (1986), no. 2, 83–127. MR 867116, DOI 10.1016/0167-8396(86)90016-6
Michel Fortin and Roland Glowinski, Augmented Lagrangian methods, Studies in Mathematics and its Applications, vol. 15, North-Holland Publishing Co., Amsterdam, 1983. Applications to the numerical solution of boundary value problems; Translated from the French by B. Hunt and D. C. Spicer. MR 724072
Max D. Gunzburger, Finite element methods for viscous incompressible flows, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989. A guide to theory, practice, and algorithms. MR 1017032
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
Ohannes A. Karakashian, On a Galerkin-Lagrange multiplier method for the stationary Navier-Stokes equations, SIAM J. Numer. Anal. 19 (1982), no. 5, 909–923. MR 672567, DOI 10.1137/0719066
Ming-Jun Lai and Paul Wenston, Bivariate splines for fluid flows, Comput. & Fluids 33 (2004), no. 8, 1047–1073. MR 2048634, DOI 10.1016/j.compfluid.2003.10.003
James Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187–195. MR 136885, DOI 10.1007/BF00253344
Roger Temam, Navier-Stokes equations, 3rd ed., Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. Theory and numerical analysis; With an appendix by F. Thomasset. MR 769654
Morten M. T. Wang and Tony W. H. Sheu, An element-by-element BICGSTAB iterative method for three-dimensional steady Navier-Stokes equations, J. Comput. Appl. Math. 79 (1997), no. 1, 147–165. MR 1437975, DOI 10.1016/S0377-0427(96)00172-0
Alexander Ženíšek, Polynomial approximation on tetrahedrons in the finite element method, J. Approximation Theory 7 (1973), 334–351. MR 350260, DOI 10.1016/0021-9045(73)90036-1