Mixed hp - finite element methods for viscous incompressible fluid flow

The efficient numerical approximation of viscous, incompressible flow by families of mixed finite elements is subject to the satisfaction of a stability or inf-sup condition between the velocity and pressure approximation spaces.;The present work analyses the stability of mixed hp-finite elements for planar Stokes flow on affine quadrilateral meshes comprising of regular and anisotropic elements. Firstly, a new family of mixed hp-finite elements is presented for regular elements with an inf-sup constant bounded below independently of the mesh size h and the spectral order p. In particular, the element allows continuous piecewise polynomial pressures to be used.;Next, the stability of families of mixed hp-finite elements on geometrically refined anisotropic elements is considered using a macro-element technique. New results are presented for edge and corner macro-elements, in particular, for the latter case, the dependence of the inf-sup constant on the geometric refinement parameter is explicitly characterised.;The families are then used in the numerical approximation of two physical Navier-Stokes problems.