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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.

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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