On stabilized finite element methods based on the Scott-Zhang projector: circumventing the inf-sup condition for the Stokes problem

Abstract

In this work we propose a stabilized finite element method that permits us to circumvent discrete inf–sup conditions, e.g. allowing equal order interpolation. The type of method we propose belongs to the family of symmetric stabilization techniques, which are based on the introduction of additional terms that penalize the difference between some quantities, i.e. the pressure gradient in the Stokes problem, and their finite element projections. The key feature of the formulation we propose is the definition of the projection to be used, a non-standard Scott–Zhang projector that is well-defined for $L^{1}(\Omega )$functions. The resulting method has some appealing features: the projector is local and nested meshes or enriched spaces are not required.