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Analysis and Convergence of a Covolume Method for the Generalized Stokes Problem


Author: S. H. Chou
Journal: Math. Comp. 66 (1997), 85-104
MSC (1991): Primary 65N15, 65N30, 76D07; Secondary 35B45, 35J50
DOI: https://doi.org/10.1090/S0025-5718-97-00792-8
MathSciNet review: 1372003
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Abstract: We introduce a covolume or MAC-like method for approximating the generalized Stokes problem. Two grids are needed in the discretization; a triangular one for the continuity equation and a quadrilateral one for the momentum equation. The velocity is approximated using nonconforming piecewise linears and the pressure piecewise constants. Error in the $L^2$ norm for the pressure and error in a mesh dependent $H^1$ norm as well as in the $L^2$ norm for the velocity are shown to be of first order, provided that the exact velocity is in $H^2$ and the true pressure in $H^1$. We also introduce the concept of a network model into the discretized linear system so that an efficient pressure-recovering technique can be used to simplify a great deal the computational work involved in the augmented Lagrangian method. Given is a very general decomposition condition under which this technique is applicable to other fluid problems that can be formulated as a saddle-point problem.


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  • 1. R. Amit, C. A. Hall and T. A. Porsching, An application of network theory to the solution of implicit Navier-Stokes difference equations, J. Compt. Phys. 40 (1981), 183-201. MR 84d:76016
  • 2. F. Brezzi and M. Fortin, Mixed and hybrid finite elements, Springer-Verlag, New York (1991). MR 92d:65187.
  • 3. S. C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Springer-Verlag, 1994. MR 95f:65001
  • 4. S. H. Chou, A network model for incompressible two-fluid flow and its numerical solution, Numer. Meth. Partial Diff. Eqns. 5 (1989), 1-24. MR 90i:76142
  • 5. -, A network model for two-fluid flow, Proceedings of the 5th International Conference on Reactor Thermal Hydraulics, American Nuclear Society, Vol. VI, Salt Lake City, Utah, 1992, pp. 1607-1614.
  • 6. S. Choudhury and R. A. Nicolaides, Discretization of incompressible vorticity-velocity equations on triangular meshes, Internat. J. Numer. Methods Fluid Dynamics 11 (1990).
  • 7. M. Crouzeix and P. A. Raviart, Conforming and nonconforming finite element methods for solving the stationary stokes equations, RAIRO Anal. Numer. 7, (1973), 33-76. MR 49:8401
  • 8. M, Fortin and R. Glowinski, Augmented Lagrangian methods: applications to the numerical solution of boundary-value problems, North-Holland, New York, 1983. MR 85a:49004
  • 9. L. Gastaldi and R. Nochetto, Optimal $L^\infty $-error estimates for nonconforming and mixed finite element methods of lowest order, Numer. Math. 50 (1987), 587-611. MR 88f:65196
  • 10. V. Girault and P. A. Raviart, Finite element methods for Navier-Stokes equations, Springer-Verlag, Berlin, and New York, 1986. MR 88b:65129
  • 11. R. Glowinski and P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM, Philadelphia, PA, 1989. MR 91f:73038
  • 12. C. A. Hall, J. C. Cavendish and W. H. Frey, The dual variable method for solving fluid flow difference equations on Delaunay triangulations, Comput. & Fluids 20 (1991), 145-164. MR 92g:76059
  • 13. C. A. Hall, T. A. Porsching and G. L. Mesina, On a network method for unsteady incompressible fluid flow on triangular grids, Internat. J. Numer. Methods Fluids 15 (1992), 1383-1406.
  • 14. F. H. Harlow and F. E. Welch, Numerical calculations of time dependent viscous incompressible flow of fluid with a free surface, Phys. Fluids 8 (1965), 2181.
  • 15. R. A. Nicolaides, Direct discretization of planar div-curl problems, SIAM J. Numer. Anal. 29 (1992), 32-56. MR 93b:65176
  • 16. R. A. Nicolaides, Analysis and convergence of the MAC scheme, SIAM J. Numer. Anal. 29 (1992), 1579-1591. MR 93j:65143
  • 17. R. A. Nicolaides, T. A. Porsching and C. A. Hall, Covolume methods in computational fluid dynamics, Computational Fluid Dynamics Review (M. Hafez and K. Oshma, eds.), Wiley, New York, 1995, pp. 279-299.
  • 18. T. A. Porsching, Error estimates for MAC-like approximations to the linear Navier-Stokes equations, Numer. Math. 29 (1978), 291-306. MR 57:11348
  • 19. -, A network model for two-fluid flow, Numer. Methods Partial Differential Equations 1 (1985), 295-313.
  • 20. G. Strang, Introduction to applied mathematics, Wellesley-Cambridge Press, Wellesley, MA, 1986. MR 88a:00006

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Additional Information

S. H. Chou
Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43402-0221
Email: chou@zeus.bgsu.edu

DOI: https://doi.org/10.1090/S0025-5718-97-00792-8
Keywords: Covolume methods, augmented Lagrangian method, nonconforming mixed finite element, network models
Received by editor(s): September 11, 1995
Received by editor(s) in revised form: December 1, 1995
Article copyright: © Copyright 1997 American Mathematical Society

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