Bivariate quadratic finite elements and vertex splines

Authors:
Charles K. Chui and Tian Xiao He

Journal:
Math. Comp. **54** (1990), 169-187

MSC:
Primary 65D07; Secondary 41A15, 41A63, 65N30

MathSciNet review:
993926

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Abstract: Following work of Heindl and of Powell and Sabin, each triangle of an arbitrary (regular) triangulation of a polygonal region in is subdivided into twelve triangles, using the three medians, yielding the refinement of , so that quadratic finite elements can be constructed. In this paper, we derive the Bézier nets of these elements in terms of the parameters that describe function and first partial derivative values at the vertices and values of the normal derivatives at the midpoints of the edges of . Consequently, bivariate quadratic (generalized) vertex splines on have an explicit formulation. Here, a generalized vertex spline is one which is a piecewise polynomial on the refined grid partition and has support that contains at most one vertex of the original partition in its interior. The collection of all quadratic generalized vertex splines on so constructed is shown to form a basis of , the vector space of all functions on whose restrictions to each triangular cell of the partition are quadratic polynomials. A subspace with the basis given by appropriately chosen generalized vertex splines with exactly one vertex of in the interior of their supports, that reproduces all quadratic polynomials, is identified, and hence, has approximation order three. Quasi-interpolation formulas using this subspace are obtained. In addition, a constructive procedure that yields a locally supported basis of yet another subspace with dimension given by the number of vertices of , that has approximation order three, is given.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1990-0993926-3

Keywords:
Bivariate splines,
interpolation,
quasi-interpolation,
macroelements,
vertex splines

Article copyright:
© Copyright 1990
American Mathematical Society