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

Author: S. H. Chou
Journal: Math. Comp. 66 (1997), 85-104
MSC (1991): Primary 65N15, 65N30, 76D07; Secondary 35B45, 35J50
DOI: https://doi.org/10.1090/S0025-5718-97-00792-8
MathSciNet review: 1372003
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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.

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