Trivariate spline approximations of 3D Navier-Stokes equations
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- by Gerard Awanou and Ming-Jun Lai PDF
- Math. Comp. 74 (2005), 585-601 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
Additional Information
- Gerard Awanou
- Affiliation: The Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455-0436
- MR Author ID: 700956
- Email: awanou@ima.umn.edu
- Ming-Jun Lai
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- Email: mjlai@math.uga.edu
- 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.
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 585-601
- MSC (2000): Primary 65D07, 65D15, 35Q30, 76D05
- DOI: https://doi.org/10.1090/S0025-5718-04-01715-6
- MathSciNet review: 2114639