A Finite Element Approximation of the Stokes Equations (original) (raw)
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Mixed finite element methods for incompressible flow: Stationary Stokes equations
Numerical Methods for Partial Differential Equations, 2009
, the authors developed and analyzed a mixed finite element method for the stationary Stokes equations based on the pseudostress-velocity formulation. The pseudostress and the velocity are approximated by a stable pair of finite elements: Raviart-Thomas elements of index k ≥ 0 and discontinuous piecewise polynomials of degree k ≥ 0, respectively. This paper extends the method to the stationary, incompressible Navier-Stokes equations. Under appropriate assumptions, we show that the pseudostress-velocity formulation of the Navier-Stokes equation and its discrete counterpart have branches of nonsingular solutions, and error estimates of the mixed finite element approximations are established as well. These equations are based on the respective physical principles: the constitutive law, the balance of linear momentum, and the conservation of mass. A tremendous amount of work has been done over many years on the computation of incompressible Navier-Stokes equations based on the velocity-pressure formulation in (1.1) or its variants (see, e.g, the books by Girault and Raviart [17], Pironneau [23], Gunzburger , and the references therein). However, the practical need of the stress tensor coupled with a rising interest in non-Newtonian flows have motivated extensive studies of mixed finite element methods in the stress-velocity-pressure formulation (1.4). There are at least two major advantages of this formulation. First, it provides a unified framework for both the Newtonian and the non-Newtonian flows. It has also been pointed out in [20] that an accurate and efficient numerical scheme for Newtonian flows under formulation (1.4) is necessary for the successful computation of non-Newtonian flows. Another advantage is that a physical quantity such as the stress is computed directly instead of by taking derivatives of the velocity. This avoids degrading the accuracy, which is inevitable in the process of numerical differentiation. However, the stress-velocity-pressure formulation has some obvious disadvantages. The most significant ones are the increase in the number of unknowns and the symmetry requirement for the stress tensor . Both of them pose extra difficulty in the numerical computation. In order to keep the pros and improve the cons of the stress-velocity-pressure formulation, Cai et al. recently analyzed and implemented mixed finite element methods based on the pseudostress-velocity formulation for the stationary Stokes equations in . Raviart-Thomas (RT) elements of index k ≥ 0 [25] are used for approximating each row of the pseudostress which is a nonsymmetric tensor, and discontinuous piecewise polynomials of degree k ≥ 0 for approximating each component of the velocity. It is shown that this pair of mixed finite elements is stable and yields optimal accuracy O(h k+1 ) in the H(div) and L 2 norms for the respective pseudostress and velocity. For lower order elements, the total number of degrees of freedom for the discretization is comparable to that for the velocity-pressure formulation using Crouzeix-Raviart elements (nonconforming velocity and discontinuous pressure), and both approaches have the same accuracy for the H 1 seminorm of the velocity and the L 2 norm of the pressure. The indefinite system of linear equations resulting from the discretization is decoupled by the penalty method. The penalized pseudostress system is solved by the H(div) type of multigrid method, and the velocity is then calculated explicitly/locally. It is shown theoretically in [11] that the convergence rate of the V (1, 1)-cycle multigrid is independent of the mesh size, the number of levels, and the penalty parameter. This is also confirmed numerically in with remarkably fast convergence rates around 0.21 for the RT element of index zero and 0.14 for the BDM element of index one . The purpose of this paper is to extend and analyze the method for the stationary Navier-Stokes equations. Numerical studies of the method for several benchmark test problems, such as the driven cavity problem and flows past cylinder, are reported in . The extension of the method is rather straightforward; the only modification needed is to replace ∇u in the nonlinear term by the deviatoric pseudostress since the velocity u is approximated by discontinuous piecewise polynomials. But the analysis is nontrivial. First, it is not known that the general theory on the well-posedness of the saddle point problem (see, e.g., [6]) can be extended for problems with the linear or nonlinear convection term. So we prove the well-posedness of the pseudostressvelocity formulation for the Navier-Stokes equations via establishing its equivalence
A new finite-element discretization of the Stokes problem coupled with the Darcy equations
IMA Journal of Numerical Analysis, 2008
The flow in a rigid porous medium with a crack is usually modeled by Darcy equations coupled with the Stokes problem. We first propose a new variational formulation of the Stokes system, where the unknowns are the vorticity, the velocity and the pressure, and describe the corresponding finite element discretization. We extend this discretization to the case where Darcy and Stokes equations are coupled and prove optimal a priori and a posteriori error estimates. We conclude with some numerical experiments. Résumé: Un modèle usuel pour l'écoulement dans un milieu poreux rigide avec une fracture consisteà coupler leséquations de Darcy et de Stokes. Nous proposons tout d'abord une nouvelle formulation variationnelle deséquations de Stokes où les inconnues sont le tourbillon, la vitesse et la pression et décrivons la discrétisation paréléments finis correspondante. Nousétendons cette discrétisation au cas où les systèmes de Darcy et de Stokes sont couplés et prouvons des estimations d'erreur a priori et a posteriori optimales. Nous présentons quelques expériences numériques.