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**Example text**

We have yet to establish property (5); the other 36 RATIONAL FINITE ELEMENT BASIS five properties in Section 1 . 5 are obviously satisfied. Let u(x,y) be a linear function with values ui at the quadrilateral vertices. Then g(x,y) = U(X,Y) - c UiWi(X,Y) i=1 4 vanishes on the perimeter of the quadrilateral by virtue of properties ( 2 ) - ( 4 ) . There must be a P2 such that g(x,y) = P2 (x,y)/Q,(x,y), where P2 is zero on the quartic perimeter. 1, this is possible only if P2 is the zero polynomial.

1) is linear on (4;l) only when (2;3) is parallel to (4;l) and on 32 R A T I O N A L FINITE ELEMENT BASIS (1;2) only when (3;4) is parallel to (1;2). Both of these conditions are met only for a parallelogram. 5 is thus violated. Property ( 4 ) implies that both (2;3) and (3;4) must appear as factors in any wedge for vertex 1. Introduction of other factors will only increase the degree of variation on the adjacent sides. 5 exists for the quadilateral, except for the special case of a parallelogram.

The order of a polypol is the sum of the orders of its sides. This is equal to the order of its boundary. An n-sided polypol is called an 'In-pol". Let side Pi of a given n-pol be of order si. 20b) 3 The value of r classifies the element: r=l is a polygon, r=2 is a polycon, and r>2 is a polypol. The value of m is the order of the element. Generalization to higher dimensions requires delicate considerations. In Chapter 7 we develop a theory for construction of basis functions for three- 29 PATCHWORK APPROXIMATION dimensional algebraic elements.