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.
 1.
R.
Amit, C.
A. Hall, and T.
A. Porsching, An application of network theory to the solution of
implicit NavierStokes difference equations, J. Comput. Phys.
40 (1981), no. 1, 183–201. MR 611808
(84d:76016), http://dx.doi.org/10.1016/00219991(81)902060
 2.
Franco
Brezzi and Michel
Fortin, Mixed and hybrid finite element methods, Springer
Series in Computational Mathematics, vol. 15, SpringerVerlag, New
York, 1991. MR
1115205 (92d:65187)
 3.
Susanne
C. Brenner and L.
Ridgway Scott, The mathematical theory of finite element
methods, Texts in Applied Mathematics, vol. 15, SpringerVerlag,
New York, 1994. MR 1278258
(95f:65001)
 4.
S.
H. Chou, A network model for incompressible twofluid flow and its
numerical solution, Numer. Methods Partial Differential Equations
5 (1989), no. 1, 1–24. MR 1012225
(90i:76142), http://dx.doi.org/10.1002/num.1690050102
 5.
, 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.
 6.
S. Choudhury and R. A. Nicolaides, Discretization of incompressible vorticityvelocity 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. I, Rev. Française
Automat. Informat. Recherche Opérationnelle Sér. Rouge
7 (1973), no. R3, 33–75. MR 0343661
(49 #8401)
 8.
Michel
Fortin and Roland
Glowinski, Augmented Lagrangian methods, Studies in
Mathematics and its Applications, vol. 15, NorthHolland 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
(85a:49004)
 9.
Lucia
Gastaldi and Ricardo
Nochetto, Optimal 𝐿^{∞}error estimates for
nonconforming and mixed finite element methods of lowest order, Numer.
Math. 50 (1987), no. 5, 587–611. MR 880337
(88f:65196), http://dx.doi.org/10.1007/BF01408578
 10.
Vivette
Girault and PierreArnaud
Raviart, Finite element methods for NavierStokes equations,
Springer Series in Computational Mathematics, vol. 5, SpringerVerlag,
Berlin, 1986. Theory and algorithms. MR 851383
(88b:65129)
 11.
Roland
Glowinski and Patrick
Le Tallec, Augmented Lagrangian and operatorsplitting methods in
nonlinear mechanics, SIAM Studies in Applied Mathematics, vol. 9,
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA,
1989. MR
1060954 (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), no. 2, 145–164. MR 1123813
(92g:76059), http://dx.doi.org/10.1016/00457930(91)90017C
 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), 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.
 15.
R.
A. Nicolaides, Direct discretization of planar divcurl
problems, SIAM J. Numer. Anal. 29 (1992), no. 1,
32–56. MR
1149083 (93b:65176), http://dx.doi.org/10.1137/0729003
 16.
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
(93j:65143), http://dx.doi.org/10.1137/0729091
 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. 279299.
 18.
T.
A. Porsching, Error estimates for MAClike approximations to the
linear NavierStokes equations, Numer. Math. 29
(1977/78), no. 3, 291–306. MR 0471622
(57 #11348)
 19.
, A network model for twofluid flow, Numer. Methods Partial Differential Equations 1 (1985), 295313.
 20.
Gilbert
Strang, Introduction to applied mathematics,
WellesleyCambridge Press, Wellesley, MA, 1986. MR 870634
(88a:00006)
 1.
 R. Amit, C. A. Hall and T. A. Porsching, An application of network theory to the solution of implicit NavierStokes difference equations, J. Compt. Phys. 40 (1981), 183201. MR 84d:76016
 2.
 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
 4.
 S. H. Chou, A network model for incompressible twofluid flow and its numerical solution, Numer. Meth. Partial Diff. Eqns. 5 (1989), 124. MR 90i:76142
 5.
 , 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.
 6.
 S. Choudhury and R. A. Nicolaides, Discretization of incompressible vorticityvelocity 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), 3376. MR 49:8401
 8.
 M, Fortin and R. Glowinski, Augmented Lagrangian methods: applications to the numerical solution of boundaryvalue problems, NorthHolland, New York, 1983. MR 85a:49004
 9.
 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
 10.
 V. Girault and P. A. Raviart, Finite element methods for NavierStokes equations, SpringerVerlag, Berlin, and New York, 1986. MR 88b:65129
 11.
 R. Glowinski and P. Le Tallec, Augmented Lagrangian and operatorsplitting 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), 145164. 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), 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.
 15.
 R. A. Nicolaides, Direct discretization of planar divcurl problems, SIAM J. Numer. Anal. 29 (1992), 3256. MR 93b:65176
 16.
 R. A. Nicolaides, Analysis and convergence of the MAC scheme, SIAM J. Numer. Anal. 29 (1992), 15791591. 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. 279299.
 18.
 T. A. Porsching, Error estimates for MAClike approximations to the linear NavierStokes equations, Numer. Math. 29 (1978), 291306. MR 57:11348
 19.
 , 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
