TABLE OF CONTENTS For the treatment of second order elliptic problems on domains with curved boundaries, we need to use elements with curved sides if we want to get higher accuracy. For many problems of fourth order, we even have to do a good job of approximating the boundary in the C 1-norm just to get convergence. For this reason, certain so-called isoparametric families of finite elements were developed. They are a generalization of the affine families.
For triangulations, isoparametric elements actually play a role only near the boundary. On the other hand, (simple) isoparametric quadrilaterals are often used in the interior since this allows us to generate arbitrary quadrilaterals, rather than just parallelograms.
We restrict our attention to planar domains, and consider families of elements where every T ? 
is generated by a bijective mapping F:
This framework includes the affine families when p and q are required to be linear functions. When p and q are polynomials of higher degree, we get the more general situation of isoparametric elements. More precisely, the polynomials in the parametrization are chosen from the same family ? as the shape functions of the element (T, ?, ?).
The important case where p and q are quadratic polynomials is shown in Fig. 31. By Remark II.5.4, we know that six points P i ,1 ? i ?6, can be prescribed. Then p
