Title: Approximation by quadrilateral finite elements

Authors: Douglas N. Arnold, Daniele Boffi, and Richard S. Falk

Source: Math. Comp., 71(239):909-922, 2002

Abstract:    We consider the approximation properties of finite element spaces
    on quadrilateral meshes. The finite element spaces are constructed
    starting with a given finite dimensional space of functions on a
    square reference element, which is then transformed to a space of
    functions on each convex quadrilateral element via a bilinear
    isomorphism of the square onto the element. It is known that for
    affine isomorphisms, a necessary and sufficient condition for
    approximation of order r+1 in L2 and order r in H1 is that 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, it is known that the same estimates hold if
    the function space contains all polynomial functions of separate
    degree r. We show, by means of a counterexample, that this latter
    condition is also necessary. As applications we demonstrate
    degradation of the convergence order on quadrilateral meshes as
    compared to rectangular meshes for serendipity finite elements and
    for various mixed and nonconforming finite elements.

Keywords: quadrilateral, finite element, approximation, serendipity,
          mixed finite element

URL: http://www.math.rutgers.edu/~falk/papers/quadapprox.pdf