Mixed $hp$ finite element methods for problems in elasticity and Stokes flow

Summary.

We consider the mixed formulation for the elasticity problem and the limiting Stokes problem in ${\Bbb R}^d$,$d=2,3$ . We derive a set of sufficient conditions under which families of mixed finite element spaces are simultaneously stable with respect to the mesh size$h$ and, subject to a maximum loss of$O (k^{\frac{d-1}{2}})$ , with respect to the polynomial degree $k$. We obtain asymptotic rates of convergence that are optimal up to$O (k^\epsilon)$ in the displacement/velocity and up to$O (k^{\frac{d-1}{2}+\epsilon})$ in the "pressure", with$\epsilon >0$ arbitrary (both rates being optimal with respect to$h$ ). Several choices of elements are discussed with reference to properties desirable in the context of the $hp$-version.