Quadrilateral finite elements of FVS type and class $C^\rho$

Abstract

Let \({\cal P}\) be some partition of a planar polygonal domain \(\Omega\) into quadrilaterals. Given a smooth function \(u\), we construct piecewise polynomial functions\(\upsilon \in C^\rho (\Omega)\) of degree \(n = 3\rho\) for\(\rho\) odd, and \(n = 3\rho +1\) for \(\rho\) even on a subtriangulation \(\tau_4\) of \({\cal P}\). The latter is obtained by drawing diagonals in each \(Q\in {\cal P}\), and \(\upsilon \vert Q\) is a composite quadrilateral finite element generalizing the classical \(C^1\) cubic Fraeijs de Veubeke and Sander (or FVS) quadrilateral. The function \(\upsilon\) interpolates the derivatives of\(u\) up to order \(\rho + [\rho/2]\) at the vertices of \({\cal P}\). Polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements.