Analysis and Convergence of a Covolume Method for the Generalized Stokes Problem
Author:
S. H. Chou
Journal:
Math. Comp. 66 (1997), 85104
MSC (1991):
Primary 65N15, 65N30, 76D07; Secondary 35B45, 35J50
MathSciNet review:
1372003
Fulltext PDF Free Access
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Additional Information
Abstract: We introduce a covolume or MAClike 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 norm for the pressure and error in a mesh dependent norm as well as in the norm for the velocity are shown to be of first order, provided that the exact velocity is in and the true pressure in . We also introduce the concept of a network model into the discretized linear system so that an efficient pressurerecovering 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 saddlepoint problem.
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 F. Brezzi and M. Fortin, Mixed and hybrid finite elements, SpringerVerlag, New York (1991). MR 92d:65187.
 3.
 S. C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, SpringerVerlag, 1994. MR 95f:65001
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 , A network model for twofluid flow, Proceedings of the 5th International Conference on Reactor Thermal Hydraulics, American Nuclear Society, Vol. VI, Salt Lake City, Utah, 1992, pp. 16071614.
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 S. Choudhury and R. A. Nicolaides, Discretization of incompressible vorticityvelocity equations on triangular meshes, Internat. J. Numer. Methods Fluid Dynamics 11 (1990).
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 M, Fortin and R. Glowinski, Augmented Lagrangian methods: applications to the numerical solution of boundaryvalue problems, NorthHolland, New York, 1983. MR 85a:49004
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 L. Gastaldi and R. Nochetto, Optimal error estimates for nonconforming and mixed finite element methods of lowest order, Numer. Math. 50 (1987), 587611. MR 88f:65196
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 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), 13831406.
 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.
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 R. A. Nicolaides, Direct discretization of planar divcurl problems, SIAM J. Numer. Anal. 29 (1992), 3256. MR 93b:65176
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 T. A. Porsching, Error estimates for MAClike approximations to the linear NavierStokes equations, Numer. Math. 29 (1978), 291306. MR 57:11348
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 , A network model for twofluid flow, Numer. Methods Partial Differential Equations 1 (1985), 295313.
 20.
 G. Strang, Introduction to applied mathematics, WellesleyCambridge 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 434020221
Email:
chou@zeus.bgsu.edu
DOI:
http://dx.doi.org/10.1090/S0025571897007928
PII:
S 00255718(97)007928
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
