Inf-Sup Stability Of Isogeometric Taylor-Hood And Sub-Grid Methods For The Stokes Problem With Hierarchical Splines

In this article, we prove the inf-sup stability of an adaptive isogeometric discretization of the Stokes problem. The discretization is based on the hierarchical generalization of the isogeometric Taylor-Hood and Sub-Grid elements, which were described by Bressan Sangalli (2013, Isogeometric discretizations of the Stokes problem: stability analysis by the macroelement technique. IMA J. Numer. Anal., 33, 629651) for tensor-product splines. In order to extend the existing proof to the hierarchical setting, we need to adapt some of the steps considerably. In particular, the required local approximation estimate is obtained by analysing the properties of the quasi-interpolant of Speleers & Manni (2016, Effortless quasi-interpolation in hierarchical spaces. Numer. Math., 132, 155-184) with respect to certain Sobolev norms.In addition to the theoretical results, we also perform numerical tests in order to analyse the dependency of the inf-sup constant on the mesh regularity assumptions. Finally, the article also presents a numerical convergence test of the resulting adaptive method on a T-shaped domain.