The stability of mixed \(hp\)-finite element methods for Stokes flow on high aspect ratio elements (Q2706387)

The authors discuss applications of finite element methods to stationary Stokes problem in polygonal domains. They pay special attention to the study of constant \(c_k\) in inf-sup condition when anisotropic meshes and polynomials of high degree \(k\) are used. This is of importance for practical solution of Navier-Stokes equations when localized features are required. Mainly quadrilateral elements are investigated on the base of special edge and corner macroelements theory. Under certain conditions on the subspaces, it is shown that \(c_k\geq c(1+\log ^{1/2}k)^{-1}\) uniformly in \(k\) and in the aspect ratio of elements. Very interesting numerical examples are presented. It might be mentioned that also methods on composite triangular grids, not considered in the paper, deserve a special attention when one deals with problems under consideration. This is especially useful in cases with complex geometry and when the construction of effective iterative methods should be taken into account.