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Volume 15,     Number 4,     Winter 2007

 

REFINED MIXED FINITE ELEMENT
METHOD FOR THE STOKES PROBLEM
HASNA EL SOSSA AND LUC PAQUET

Abstract. We consider the Dirichlet problem for the Stokes problem in a plane polygonal domain (simply connected) with a reentrant corner at the origin. In addition to the velocity field , the tensor field is introduced as a supplementary unknown leading us to a mixed formulation. Note that contrarily to [4], that our mixed formulation of the continuous problem is not hybridized. To discretize the mixed formulation, given a fixed triangulation, each of the two lines of the tensor field σ is approximated on each triangle of the given triangulation by a Raviart-Thomas vector field of degree 0 and by a constant vector field. Under suitable refinement conditions on the regular family of triangulations under consideration , we prove that despite to the reentrant corner, that the convergence of the sequence of approximate solutions to ((σ,p),) is still of order 1 in the L2-norm. Finally, the discretized mixed formulation is hybridized by introducing a Lagrange multiplier to relax the continuity of the normal trace of σh across interelement edges eh of the triangulation Th. A formula expressing in terms of h and σh is established. Under some additional assumptions, it is proved that uniformly in eh, when h → o+. Without these additional assumptions, the behaviour of the {h} at the boundary Γ is studied. The jumps of h as one crosses from one triangle to an adjacent one are studied and also compared with those of the means of on each triangle of the triangulation Th. In the late section about the numerical implementation, we first derive explicit formulas for σ h|K and h|K on each triangle K of the triangulation Th in terms of the Lagrange multiplier . We then reduce our problem to the resolution of a linear system with explicit coefficients in terms of the geometry of the triangulation with only the Lagrange multiplier on each edge and the discrete pressure ph on each triangle as unknowns.

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© 2007, Canadian Applied Mathematics Quarterly (CAMQ)