Analysis and convergence of a covolume method for the generalized Stokes problem

Chou, S.

doi:10.1090/S0025-5718-97-00792-8

Analysis and convergence of a covolume method for the generalized Stokes problem
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by S. H. Chou PDF

Math. Comp. 66 (1997), 85-104 Request permission

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.

References

R. Amit, C. A. Hall, and T. A. Porsching, An application of network theory to the solution of implicit Navier-Stokes difference equations, J. Comput. Phys. 40 (1981), no. 1, 183–201. MR 611808, DOI 10.1016/0021-9991(81)90206-0
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
S. H. Chou, A network model for incompressible two-fluid flow and its numerical solution, Numer. Methods Partial Differential Equations 5 (1989), no. 1, 1–24. MR 1012225, DOI 10.1002/num.1690050102
—, 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.
S. Choudhury and R. A. Nicolaides, Discretization of incompressible vorticity-velocity equations on triangular meshes, Internat. J. Numer. Methods Fluid Dynamics 11 (1990).
M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. MR 343661
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
Lucia Gastaldi and Ricardo Nochetto, Optimal $L^\infty$-error estimates for nonconforming and mixed finite element methods of lowest order, Numer. Math. 50 (1987), no. 5, 587–611. MR 880337, DOI 10.1007/BF01408578
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
Roland Glowinski and Patrick Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM Studies in Applied Mathematics, vol. 9, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1060954, DOI 10.1137/1.9781611970838
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), no. 2, 145–164. MR 1123813, DOI 10.1016/0045-7930(91)90017-C
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.
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.
R. A. Nicolaides, Direct discretization of planar div-curl problems, SIAM J. Numer. Anal. 29 (1992), no. 1, 32–56. MR 1149083, DOI 10.1137/0729003
R. A. Nicolaides, Analysis and convergence of the MAC scheme. I. The linear problem, SIAM J. Numer. Anal. 29 (1992), no. 6, 1579–1591. MR 1191137, DOI 10.1137/0729091
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.
T. A. Porsching, Error estimates for MAC-like approximations to the linear Navier-Stokes equations, Numer. Math. 29 (1977/78), no. 3, 291–306. MR 471622, DOI 10.1007/BF01389214
—, A network model for two-fluid flow, Numer. Methods Partial Differential Equations 1 (1985), 295–313.
Gilbert Strang, Introduction to applied mathematics, Wellesley-Cambridge Press, Wellesley, MA, 1986. MR 870634