Vorticity-velocity-pressure formulation for Stokes problem

Abstract

Abstract. We propose a three-field formulation for efficiently solving a twodimensional Stokes problem in the case of nonstandard boundary conditions. More specifically, we consider the case where the pressure and either normal or tangential components of the velocity are prescribed at some given parts of the boundary. The proposed computational methodology consists in reformulating the considered boundary value problem via a mixed-type formulation where the pressure and the vorticity are the principal unknowns while the velocity is the Lagrange multiplier. The obtained formulation is then discretized and a convergence analysis is performed. A priori error estimates are established, and some numerical results are presented to highlight the perfomance of the proposed computational methodology.

Stabilized finite element method for Navier--Stokes equations with physical boundary conditions

“…We thus obtain existence and uniqueness of the solution of the discrete problem in a neighbourhood of the exact solution, unconditional convergence of the approximation of the Navier-Stokes equations and also a priori and a posteriori error estimates for the vorticity and the pressure in L 2 (Ω), respectively for the velocity in L 4 (Ω). For smooth solutions, one gets the same convergence rate O(h) as for the Stokes problem.…”

“…The corresponding 2D Stokes equations with the same boundary conditions were studied in [12] and [4], by means of different three-fields variational formulations. In [4], after showing that the new vorticity-velocity-pressure formulation proposed was well-posed, the authors discretized it by means of conforming low-order finite elements.…”

“…The corresponding 2D Stokes equations with the same boundary conditions were studied in [12] and [4], by means of different three-fields variational formulations. In [4], after showing that the new vorticity-velocity-pressure formulation proposed was well-posed, the authors discretized it by means of conforming low-order finite elements. The discrete inf-sup condition is then obvious, while the discrete coercivity is obtained by adding a stabilization term (see [1]) taking into account the jumps of both the vorticity and the pressure across the edges of the triangulation.…”

“…The discretization is based on the method developed in [4] for the associated Stokes equations, the present paper being a generalization to the nonlinear case. However, we consider here a simpler discrete formulation of the Stokes equations, in which we stabilize only the pressure and which yields similar results from both theoretical and numerical points of view.…”

A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions

“…In order to discretize the Navier-Stokes equations provided with the boundary conditions on Γ m , a new formulation has been recently introduced in [20] (see also [11,12]) and has been extended to mixed conditions in [1]: the vorticity of the fluid is considered as a third independent unknown. However, we prefer to keep the formulation with two unknowns, in order to make use of standard finite element results.…”