Approximation by quadrilateral finite elements

Abstract

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 L p and order r in W 1 p 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.

Finite element differential forms on curvilinear cubic meshes and their approximation properties

“…For 0-forms, the requirement reduces to inclusion of the space Q r (K ). This result was obtained previously in [5] in two dimensions, and in [18], in three dimensions. The requirement becomes even more stringent as the form degree, k, is increased.…”

“…It follows that (24) holds if and only if s ≤ max(2, r/n + 1), which gives the L 2 rate of convergence of the serendipity elements on curvilinear meshes. This was shown in two dimensions in [5].…”

“…Thus we are led to ask what conditions on the reference shape functions V (K ) and the mappings F K ensure that this is so. Necessary and sufficient conditions on the space of reference shape functions were given for multilinearly mapped cubical elements in the case of 0-forms in two and three dimensions in [5] and [18], respectively. For the case of (n − 1)-forms, i.e., H (div) finite elements, such conditions were given in n = 2 dimensions in [6], in the lowest order case r = 1 in three dimensions in [16], and for general r in three dimensions in [9].…”

“…In such cases, the shape functions for the finite element space are defined on the square or cubic reference element and then transformed to the deformed physical element. In [5] it was shown in the case of scalar (H 1 ) finite elements in two dimensions that, depending on the choice of reference shape function space, this procedure may result in a loss of accuracy in comparison with the accuracy achieved on meshes of squares. Thus, for example, the serendipity finite element space achieves only about one half the rate of approximation when applied on general quadrilateral meshes, as compared to what it achieves on meshes of squares.…”

“…Thus, for example, the serendipity finite element space achieves only about one half the rate of approximation when applied on general quadrilateral meshes, as compared to what it achieves on meshes of squares. The results of [5] were extended to three dimensions in [18]. The case of vector (H (div)) finite elements in two dimensions was studied in [6].…”