Istituto di Matematica Applicata e Tecnologie Informatiche     
Arnold D. N., Boffi D., Falk R. S. Quadrilateral H(div) finite elements. Technical report ercim.cnr.ian//2002-1283, 2002.
We consider the approximation properties of quadrilateral finite element spaces of vector-valued functions 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-valued functions on a square reference element, which is then transformed to a vector-valued space of functions on each convex quadrilateral element via a 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. In particular, we show that for optimal order $L^2$ approximation, the space of functions on the reference element must contain the space generated by the standard basis functions for the local Raviart-Thomas space of degree $r$, but replacing the basis functions $(hat x^{r+1} hat y^r, 0)$ and $(0, hat x^r hat y^{r+1})$ by the single basis function $(hat x^{r+1} hat y^r, -hat x^r hat y^{r+1})$. Additional functions must be added to obtain optimal approximation of the 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).
Subject Quadrilateral, finite element, approximation, mixed finite element
65N30, 41A10, 41A25, 41A27, 41A63

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