The invertibility of the isoparametric mappings for triangular quadratic Lagrange finite elements

A reference triangular quadratic Lagrange finite element consists of a right triangle $\hat K$ with unit legs S ₁, S ₂, a local space $\hat L$ of quadratic polynomials on $\hat K$ and of parameters relating the values in the vertices and midpoints of sides of $\hat K$ to every function from $\hat L$ . Any isoparametric triangular quadratic Lagrange finite element is determined by an invertible isoparametric mapping ${F_h} = ({F_1},{F_2}) \in \hat L \times \hat L$ . We explicitly describe such invertible isoparametric mappings F _h for which the images F _h (S ₁), F _h (S ₂) of the segments S ₁, S ₂ are segments, too. In this way we extend the well-known result going back to W.B. Jordan, 1970, characterizing those invertible isoparametric mappings whose restrictions to the segments S ₁ and S ₂ are linear.