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Extra info for A Rational Finite Element Basis
Referring to Eqs. 121, we observe that all the linear forms appearing in the triangle and parallelogram wedges were determined by the sides of these figures. This simple observation is crucial in our search for a rational basis for approximation over quadrilaterals. We shall soon see that the quadrilateral itself reaches out to give us the desired linear form. 1. If three lines intersect at a point, then the ratio of linear forms which vanish on any two of these lines is constant on the third line.
We note that the exterior diagonal is uniquely defined as the line that intersects the sides of the quadrilateral at all the exterior intersection points of these sides and at no other points. 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 ) .
This simple observation is crucial in our search for a rational basis for approximation over quadrilaterals. We shall soon see that the quadrilateral itself reaches out to give us the desired linear form. 1. If three lines intersect at a point, then the ratio of linear forms which vanish on any two of these lines is constant on the third line. Proof. Referring to Fig. /sin B 3 [(a;k)/(c;k)l I j = sin A/sin B Fig. 2. T h r e e l i n e s m e e t i n g at a p o i n t . 34 RATIONAL FINITE ELEMENT BASIS is independent of j.