hp -Approximation Theory for BDFM and RT Finite Elements on Quadrilaterals

We study approximation properties of hp-finite element subspaces of H( div,Ω) and H( rot,Ω) on a polygonal domain Ω 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 Hωs,(Ω) and the countably normed spaces Bw^(Ω). 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 H¹(Ω) case. By way of illustration, exponential convergence rates are deduced for mixed hp-approximation of flow in porous media.