On the Inf-sup Condition for Higher Order Mixed Fem on Meshes with Hanging Nodes

We consider higher order mixed finite element methods for the incompressible Stokes or Navier-Stokes equations with Qr-elements for the velocity and discontinuous Pr−1-elements for the pressure where the order r can vary from element to element between 2 and a fixed bound r∗. We prove the inf-sup condition uniformly with respect to the meshwidth h on general quadrilateral and hexahedral meshes with hanging nodes. Mathematics Subject Classification. 65N30, 65N35. Received: October 3, 2005. Revised: August 30, 2006. Introduction The use of higher order mixed finite elements is well established for the numerical approximation of incompressible flow problems. In the last decade, many efforts have been made in that context in order to combine judiciously the h-version of the finite element method (FEM) with the p-version or spectral type methods. For properly designed meshes, the resulting hp-FEM has been shown to give exponential rates of convergence even in the presence of singularities (see e.g. [2, 21]). Typically the solution of incompressible Stokes or Navier-Stokes problems exhibits such singularities in the neighborhood of re-entrant corners (see e.g. [25] and references therein). Due to the incompressibility constraint ∇ · u = 0, it is well known that the pair of finite element spaces used for the approximation of velocity and pressure can not be chosen arbitrarily. A compatibility condition known as the inf-sup condition has to be satisfied in order to guarantee stability and uniqueness of the discrete solution (see e.g. [9]). Under various assumptions on the underlying meshes, the inf-sup condition has been proven for many pairs of finite element spaces both for the h-version (see e.g. [5, 8, 9, 13, 23]) and the p-version (see e.g. [3, 4, 10, 24]). However, this issue is more intricate for the case of hp-FEM which allows both local refinement by means of hanging nodes as well as various polynomial orders on the mesh elements. In the