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Extra resources for A Rational Finite Element Basis
They are easily evaluated, continuous, may be differentiated and integrated readily any number of times, and there are many results in approximation theory concerning polynomials. Rational functions, over regions bounded away from curves along which the denominators of the rational functions vanish, share many of these beneficial properties. In fact, only integration is significantly more tedious. 5. The simpleqt form for the numerator of W1 is (2;3)(3;4) [property (4)]. Thus the rational function of least degree in numerator and denominator which can be a candidate for this wedge is We therefore seek a linear form Q1 such that 33 THE QUADRILATERAL (a) Q f 0 within the quadrilateral, and 1 (b) (2;3)(3;4)/Q1 is linear on both (4;l) and (1;21.
A function with this property is said to be "regular". (6) 23 PATCHWORK APPROXIMATION Polynomial wedges satisfying these conditions exist (and are well known) for triangles and parallelograms. We will demonhtrate that for any well-set polycon rational wedges which have these properties can be found by a definitive construction of surprising simplicity. The rational wedge basis functions are not defined at points where the denominator polynomials vanish. Eq. 18) applies over the polycon where, by property (6), the wedges are well defined.
To obtain a system of projective coordinates, we first select four points, no three of which are collinear. We then choose three of these points as vertices of a "triangle of reference". The projective coordinates of these three points are equal to their barycentric coordinates. The fourth point is the "unit" point with projective coordinates defined 41 THE QUADRILATERAL as (l,l,l). The barycentric coordinates of the unit point are determined uniquely (up to a common multiplier, of course) by the location of the other three points.
A Rational Finite Element Basis by Wachspress