A uniform inf--sup condition with applications to preconditioning
For researchers working on preconditioners for parameter-dependent Stokes problems, this removes the convexity assumption, making the theory applicable to more general domains.
The paper establishes a uniform inf-sup condition for a parameter-dependent Stokes problem without requiring domain convexity, enabling uniform preconditioners. As a byproduct, a new projection operator for the Taylor-Hood element is constructed in 2D that is uniformly bounded in L2 and commutes with the divergence operator.
A uniform inf-sup condition related to a parameter dependent Stokes problem is established. Such conditions are intimately connected to the construction of uniform preconditioners for the problem, i.e., preconditioners which behave uniformly well with respect to variations in the model parameter as well as the discretization parameter. For the present model, similar results have been derived before, but only by utilizing extra regularity ensured by convexity of the domain. The purpose of this paper is to remove this artificial assumption. As a byproduct of our analysis, in the two dimensional case we also construct a new projection operator for the Taylor-Hood element which is uniformly bounded in $L^2$ and commutes with the divergence operator. This construction is based on a tight connection between a subspace of the Taylor-Hood velocity space and the lowest order Nedelec edge element.