Title: Quadrilateral H(div) finite elements.
Authors: Douglas N. Arnold, Daniele Boffi and Richard S. Falk
Source: SIAM J. Numer. Anal. 42 (2005), pp. 2429-2451
Status: Published
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 approximation.
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 L^2 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 L^2-approximation of the function
and of its divergence. As applications, we demonstrate
degradation of the convergence order on quadrilateral meshes as
compared to rectangular meshes for some standard finite element
approximations of H(div). We also derive new estimates for
approximation by quadrilateral Raviart--Thomas 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
aproximations of the solution of Poisson's equation.
Keywords: Quadrilateral, finite element, approximation, mixed finite
element
Subj. Class: 65N30, 41A10, 41A25, 41A27, 41A63
URL: http://www.math.rutgers.edu/~falk/vecquad.pdf