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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