hp-Approximation Theory forBDFMandRTFinite Elements on Quadrilaterals

We study approximation properties of hp-finite element subspaces of $oldsymbol{mathsf{H}}(mathop{{ m div}},Omega)$ and $oldsymbol{mathsf{H}}(mathop{{ m rot}},Omega)$ on a polygonal domain $Omega$ using Brezzi--Douglas--Fortin--Marini (BDFM) or Raviart--Thomas (RT) elements. Approximation theoretic results are derived for the hp-version finite element method on non-quasi-uniform meshes of quadrilateral elements with hanging nodes for functions belonging to weighted Sobolev spaces ${oldsymbol{mathsf{H}}}_{omega}^{s,ell}(Omega)$ and the countably normed spaces $pmb{{cal B}}_{w}^{ell}(Omega)$. These results culminate in a proof of the characteristic exponential convergence property of the hp-version finite element method on suitably designed meshes under similar conditions needed for the analysis of the ${oldsymbol{mathsf{H}}}^{1}(Omega)$ case. By way of illustration, exponential convergence rates are deduced for mixed hp-approximation of flow in porous media