Analysis and convergence of a covolume method for the generalized Stokes problem.(English)Zbl 0854.65091
Summary: 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.
MSC:
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 35B45 | A priori estimates in context of PDEs |
| 35J50 | Variational methods for elliptic systems |
Keywords:
covolume methods;augmented Lagrangian method;nonconforming mixed finite element;network modelsCite
Full Text:DOI
References:
| [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. Comput. Phys. 40 (1981), no. 1, 183 – 201. ·Zbl 0452.76024 ·doi:10.1016/0021-9991(81)90206-0 |
| [2] | Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. ·Zbl 0788.73002 |
| [3] | 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. ·Zbl 0804.65101 |
| [4] | 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. ·Zbl 0669.76130 ·doi:10.1002/num.1690050102 |
| [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. ·Zbl 0825.94013 |
| [6] | S. Choudhury and R. A. Nicolaides, Discretization of incompressible vorticity-velocity equations on triangular meshes, Internat. J. Numer. Methods Fluid Dynamics 11 (1990). ·Zbl 0704.76016 |
| [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. R-3, 33 – 75. ·Zbl 0302.65087 |
| [8] | 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. ·Zbl 0525.65045 |
| [9] | Lucia Gastaldi and Ricardo Nochetto, Optimal \?^{\infty }-error estimates for nonconforming and mixed finite element methods of lowest order, Numer. Math. 50 (1987), no. 5, 587 – 611. ·Zbl 0597.65080 ·doi:10.1007/BF01408578 |
| [10] | 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. ·Zbl 0585.65077 |
| [11] | 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. ·Zbl 0698.73001 |
| [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. ·Zbl 0729.76047 ·doi:10.1016/0045-7930(91)90017-C |
| [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. ·Zbl 0825.76445 |
| [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. ·Zbl 1180.76043 |
| [15] | R. A. Nicolaides, Direct discretization of planar div-curl problems, SIAM J. Numer. Anal. 29 (1992), no. 1, 32 – 56. ·Zbl 0745.65063 ·doi: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. ·Zbl 0764.76051 ·doi:10.1137/0729091 |
| [17] | R. A. Nicolaides, T. A. Porsching and C. A. Hall, Covolume methods in computational fluid dynamics, Computational Fluid Dynamics Review , Wiley, New York, 1995, pp. 279-299. ·Zbl 0875.76410 |
| [18] | T. A. Porsching, Error estimates for MAC-like approximations to the linear Navier-Stokes equations, Numer. Math. 29 (1977/78), no. 3, 291 – 306. ·Zbl 0352.65057 ·doi:10.1007/BF01389214 |
| [19] | -, A network model for two-fluid flow, Numer. Methods Partial Differential Equations 1 (1985), 295-313. ·Zbl 0637.76112 |
| [20] | Gilbert Strang, Introduction to applied mathematics, Wellesley-Cambridge Press, Wellesley, MA, 1986. ·Zbl 0618.00015 |
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