A dG Method for the Strain-Rate Formulation of the Stokes Problem Related with Nonconforming Finite Element Methods

We study a discontinuous Galerkin method for the Stokes problem written in terms of the strain-rate tensor. We approach the velocity by polynomials of degree k and the pressure by polynomials of degree k − 1 by element for k = 1, 2 or 3. The stabilization of the viscous term is new and involves the jump across the edges of the L 2-projection on P k − 1 of the velocity. It allows us to recover, when the stabilization parameter γ tends towards infinity, some stable and well-known nonconforming approximations; moreover, the inf-sup constant is independent of γ. This allows us to conclude that our method is robust with respect to γ. For k = 1, a second stabilization term is added in order to retrieve a discrete Korn inequality. The choice of the strain-rate formulation presents two main advantages, steming from its equivalence with a three-fields formulation of the Stokes problem. First, it can be easily extended to non-Newtonian liquids. Second, it allows us to deal with more physical boundary conditions involving the normal stress. Optimal a priori error estimates are also derived and numerical tests illustrating the accuracy and the robustness of the scheme are presented.