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Additional resources for A Rational Finite Element Basis
2 . 7 . W1(x,y) = (5 + 2~ 6 y ( 3 ~- A s a m p l e quadrilateral. - - y)/3(20 W2(x,y) = 20y(4 2~ W3(x,y) = 2y)/(20 + + 2~ W4(x,y) = 2(3~ - - + 8y) (4 - 2 ~ y)/(20 2y) ( 5 8~ - - 8~ + 8~ - 1 7 ~ ) ~ - 1 7 ~ ) ~ 1 7 ~ ) ~ 8~)/3(20 + 8~ - 17~). We verify linearity of W1 on side (1;2) where y = 7x and (4 3x/2: 20 + 8x 17y = 5(4 2) 2x y) = 7x mod (li2). Hence, Wl(x,y) = 1 + (4 - 2) % - - - 5 mod (1;Z). 5 - - 5 PROJECTIVE COORDINATES* One of the pleasing aspects of the development of rational bases is the interrelationship between the geometry of the elements and the algebra.
We have already demonstrated,however, that at least m wedges 22 RATIONAL FINITE ELEMENT BASIS are required for a polycon of order m. 5 WEDGE PROPERTIES. We now summarize properties thus far required of the wedge basis functions to achieve continuous patchwork degree one approximation over a collection of well-set polycons : (1) There is a node at each vertex and on each conic side. For each node there is an associated wedge within each polycon containing the node. ( 2 ) Wedge Wi(x,y) associated with node i is normalized to unity at node i.
A central pnoblem in convergence analysis is to bound (from above) the error 11 Uh-u [IA by an expression of the form chS Ilull,, where c is a constant, the A- and Bnorms are meaningful in the sense that they give useful error measures, and s is as large as possible for the prescribed scheme. We demand that our wedge basis functions be regular (infinitely differentiable) within their associated polycons. Two properties of approximation by linear combination of these wedges play a significant role in convergence analysis: (1) order of continuity across polycon boundaries, and (2) degree of polynomial for which the wedges form a basis within each polycon.
A Rational Finite Element Basis by Wachspress