Isoparametric finite element

An isoparametric finite element uses the same reference shape functions to interpolate its geometry and its unknown field. With reference functions $N_i$ and physical nodes ${\mathbf{x}}_i$, the geometry map is $F={\sum }_i{\mathbf{x}}_i{N}_i$ and the physical functions are $N_i\circ F^{-1}$. The map must be one-to-one and have a nonvanishing Jacobian determinant. For bilinear quadrilaterals, compatible common edges have affine geometry maps and identical linear field traces, giving a continuous conforming finite element space. On a general quadrilateral the pulled-back functions need not be polynomials in physical coordinates.