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Extra resources for A Rational Finite Element Basis
Function f is linear on curve P if there are any constants a(P) , b(P) , c(P) for which f(x,y) [a(P)x + b(P)y +c(P)] mod P . ) Hence, linearity of the patchwork approximation on a straight line side together with fitting of vertex values ensures continuity across the side. On any curved side of a polycon,however, a linear form has three degrees of freedom. Continuity is achieved by fitting the function value at 15 PATCHWORK APPROXIMATION another point on each conic side. This point is called a "side node".
Some of the definitions and symbols introduced in this chapter are new. 5 . 30 RATIONAL FINITE ELEMENT BASIS have been implicit in the finite element literature, but they have not previously been brought so sharply into focus. Recognition of the importance of these properties is a starting point for construction of finite element basis functions. Isoparametric coordinates are an ingenious alternative for a useful class of elements. Analysis by Ciarlet and Raviart (1972 a,b) provides a theoretical basis for application.
R. Mitchell while the author was a visiting fellow at the University of Dundee, which opened this entire line of analysis. 31 Chapter 2 THE QUADRILATERAL I t i s a f o u r - g o n c o n c l u s i o n t h a t the r e c t a n g l e i s a p a r a - g o n of v i r t u e . 5 have been known for many years for the triangle and the parallelogram. When we seek corresponding wedges for a quadrilateral which is not a parallelogram, we encounter insurmountable difficulties with polynomials. Referring to Fig.