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Quadrilateral finite elements of FVS type and class \(C^ \rho\). (English) Zbl 0824.41012
Summary: Let \({\mathcal P}\) be some partition of a planar polygonal domain \(\Omega\) into quadrilaterals. Given a smooth function \(u\), we construct piecewise polynomial functions \(v\in C^ p (\Omega)\) of degree \(n= 3\rho\) for \(\rho\) odd, and \(n= 3\rho+1\) for \(\rho\) even on a subtriangulation \(\tau_ 4\) of \({\mathcal P}\). The latter is obtained by drawing diagonals in each \(Q\in {\mathcal P}\), and \(v\mid Q\) is a composite quadrilateral finite element generalizing the classical \(C^ 1\) cubic Fraeijs de Veubeke and Sander (or FVS) quadrilateral. The function \(v\) interpolates the derivatives of \(u\) up to order \(\rho+ [\rho/ 2]\) at the vertices of \({\mathcal P}\). Polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements.

MSC:
41A15 Spline approximation
41A05 Interpolation in approximation theory
65D07 Numerical computation using splines
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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