Computer Methods in Applied Mechanics and Engineering
First vorticity–velocity–pressure numerical scheme for the Stokes problem
Section snippets
Statement of the problem
Let be a bounded connected domain of with an assumed regular boundary . The Stokes problem models the stationary equilibrium of an incompressible viscous fluid when the velocity u is sufficiently small in order to neglect the nonlinear terms (see e.g. [27]). From a mathematical point of view, this problem is the first step in order to consider the nonlinear Navier–Stokes equations of incompressible fluids, as proposed for example in [23]. The Stokes problem can be classically written
Numerical discretization
Let be a triangulation of the domain . For the sake of simplicity, we shall assume that is polygonal, in such a way that it is entirely covered by the mesh . Moreover, we will suppose that the trace of the triangulation on the boundary is such that the boundary edge of any triangle does not overlap different parts of the boundary, Γm and Γp on the one hand, Γθ and Γt on the other hand. Then, we denote by the set of triangles in . Definition 2 We suppose that belongs to the set ofFamily of regular meshes
Bercovier–Engelman test case (Figs. 4–7)
Numerical experiments have been performed first on a unit square with an analytical solution proposed by Bercovier and Engelman [9] (Fig. 4, Fig. 5, Fig. 6). The boundary conditions are formulated as follows:So Γθ and Γm are equal. Fig. 7 shows that the scheme is stable on a triangular mesh as announced in Theorem 17, and convergence is as expected: order 1 for the curl of the vorticity and for the velocity and more than 1 for the
Numerical experiments for Dirichlet velocity boundary conditions
Note that the results obtained in the previous section suppose that the vorticity is known on the part of the boundary where the normal velocity is also known. Now, we study the numerical behaviour of the scheme with general boundary conditions.
Conclusion
The vorticity–velocity–pressure variational formulation of the bidimensional Stokes problem for incompressible fluids was introduced in [16] with the vorticity chosen in space ( in bidimensional domains). In this paper, the well-posedness of this problem is theoretically proven for a particular case of Dirichlet vorticity boundary condition. We have here introduced a numerical discretization of the vorticity–velocity–pressure variational formulation and proven theoretically and
Acknowledgements
The authors would like to thank the referee for his very interesting remarks and Christine Bernardi for helpful discussions.
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