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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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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


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.
  • 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
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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:
  • Received by editor(s): September 11, 1995
  • Received by editor(s) in revised form: December 1, 1995
  • © Copyright 1997 American Mathematical Society
  • Journal: Math. Comp. 66 (1997), 85-104
  • MSC (1991): Primary 65N15, 65N30, 76D07; Secondary 35B45, 35J50
  • DOI:
  • MathSciNet review: 1372003