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Additional info for A Rational Finite Element Basis
We merely wish to indicate here the importance of continuity and of 19 PATCHWORK APPROXIMATION degree polynomials for which the wedges form a basis within each polycon. Regarding the latter, the approximation space is said to be of "degree k - 1" within a polycon if the wedge functions provide a basis for all polynomials of degree less than k. It has been shown(Strang and Fix, 1973) that for a wide class of problems a finite element space of degree k - 1 over each element achieves approximation of order hk to an arbitrary smooth function and of order hk-s to its derivatives of order s.
Hence, the absolute value of the Thus the absolute determinant of J2 is k4m4n4/s3 value of the Jacobian of the transformation from (x,y) to (q,r) coordinates is . 14) where K567 is the area of the triangle of reference whose vertices are the diagonal points of the quadrangle. 15) ( 4 ; l ) = c (m k-k4m), 1 (5;6) = c5k. (3;4) = c4(n4k-k4n), The C I S are normalizing constants. tion to (q,r) is The transforma- k = k41/s, m = qm4/s, n = rn4/s. 17) into Eq. 3 we obtain: s(qilri) (1 + qiq) (1 + rir) 4 s (q,r) ‘i q,r) = .
These forbidden shapes have their counterpart in our restriction to well-set polycons. In the isoparametric formulation, a node is introduced on each side (linear or curved) in addition to the vertex nodes. The (physical) curved sides are replaced by parabolas passing through the vertices and side nodes as shown in Fig. 13. TACTUAL F i g . 13 . The 3 - c o n : a c t u a l a n d model. MATI ON The local coordinate system is completely defined by the location of the six 3-con or eight 4-con nodes.
A Rational Finite Element Basis by Wachspress