First vorticity–velocity–pressure numerical scheme for the Stokes problem

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Abstract

We consider the bidimensional Stokes problem for incompressible fluids and recall the vorticity, velocity and pressure variational formulation, which was previously proposed by one of the authors, and allows very general boundary conditions. We develop a natural implementation of this numerical method and we describe in this paper the numerical results we obtain. Moreover, we prove that the low degree numerical scheme we use is stable for Dirichlet boundary conditions on the vorticity. Numerical results are in accordance with the theoretical ones. In the general case of unstructured meshes, a stability problem is present for Dirichlet boundary conditions on the velocity, exactly as in the stream function-vorticity formulation. Finally, we show on some examples that we observe numerical convergence for regular meshes or embedded ones for Dirichlet boundary conditions on the velocity.

Section snippets

Statement of the problem

Let Ω be a bounded connected domain of R2 with an assumed regular boundary Ω≡Γ. 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 T be a triangulation of the domain Ω. For the sake of simplicity, we shall assume that Ω is polygonal, in such a way that it is entirely covered by the mesh T. 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 ET the set of triangles in T.

Definition 2

Family Uσ of regular meshes

We suppose that T belongs to the set Uσ of

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:ω=256(y2(y−1)2(6x2−6x+1)+x2(x−1)2(6y2−6y+1))onΓ,u·n=0onΓ.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 H(curl,Ω) (=H1(Ω) 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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