A new nonconforming Galerkin scheme for the stokes problem: Partially circumventing the discrete Babǔska-Brezzi condition

Abstract

In this paper we introduce a new stable nonconforming Galerkin scheme for the interior Stokes problem. Our approach uses a Dirichllety mapping D from the pressure space Q: L 0 2(Ω) into the velocity space H := [H 0 1(Ω)] 2 in order to make a suitable change of variables in the original mixed formulation. This transformation yields an equivalent variational formulation whose associated bilinear form is strongly coercive on the product space H × Q. We then replace the mapping D by its finite element approximation on a subspace H h of H, and define the corresponding discrete scheme on a finite element subspace H h × Q h of H × Q. The leads to a nonconforming Galerkin method whose bilinear form is uniformly strongly coercive on any family {H h × Q h } h, h>0 , provided that h = O ( h) Consequently, although h = O( h) implies the satisfaction of the discrete Babǔska-Brezzi condition between H h and Q h , it turns out that this stability condition is not needed between the approximating spaces H h and Q h . Therefore, all the combinations of velocity and pressure subspaces with continuous Lagrange elements, including certainly equal-order interpolations, become stable. In addition, the Cea estimate is valid for all h, h>0 , and hence the usual rates of convergence for the error are also derived. Finally, we discuss the matrix formulation of the algorithm, and show that the resulting Galerkin equations reduce to two uncoupled linear systems whose stiffness matrices are positive definite.