 
Summary: QUADRILATERAL H(div) FINITE ELEMENTS
DOUGLAS N. ARNOLD, DANIELE BOFFI, AND RICHARD S. FALK§
SIAM J. NUMER. ANAL. c 2005 Society for Industrial and Applied Mathematics
Vol. 42, No. 6, pp. 24292451
Abstract. We consider the approximation properties of quadrilateral finite element spaces of
vector fields defined by the Piola transform, extending results previously obtained for scalar approx
imation. The finite element spaces are constructed starting with a given finite dimensional space
of vector fields on a square reference element, which is then transformed to a space of vector fields
on each convex quadrilateral element via the Piola transform associated to a bilinear isomorphism
of the square onto the element. For affine isomorphisms, a necessary and sufficient condition for
approximation of order r + 1 in L2 is that each component of the given space of functions on the
reference element contain all polynomial functions of total degree at most r. In the case of bilinear
isomorphisms, the situation is more complicated and we give a precise characterization of what is
needed for optimal order L2approximation of the function and of its divergence. As applications,
we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rect
angular meshes for some standard finite element approximations of H(div). We also derive new
estimates for approximation by quadrilateral RaviartThomas elements (requiring less regularity)
and propose a new quadrilateral finite element space which provides optimal order approximation in
H(div). Finally, we demonstrate the theory with numerical computations of mixed and least squares
finite element approximations of the solution of Poisson's equation.
