Let $\Omega = (0,1)^d$ and assume there is a tensor product mesh $\mathcal T_h$ of quadrilaterals, of maximum diameter $h$, covering $\overline \Omega$, i.e., $\overline{\Omega}= \bigcup_{K\in\mathcal T_h}K$.
Let $\mathbf F:\Omega \to \mathbb R^d$ be such that
- $\mathbf F|_K$ belongs to $C^\infty$;
- $\mathbf F$ is, at least, of class $C^2(\overline{\Omega})$.
Let $\overline{\mathbf F}_h$ be a multilinear piecewise nodal interplant of $\mathbf F$ on $\mathcal T_h$.
The following interpolation estimate holds: \begin{equation} \| |\det D \mathbf F| D \mathbf F^{-T}- |\det D\overline {\mathbf F}_h| D \overline {\mathbf{F}}_h^{-T}\|_{L^\infty(\Omega)} \le C h \| D\mathbf F\|^{d-1}_{L^\infty(\Omega)} \frac{\| D^2 \mathbf F\|_{L^\infty(\Omega)}}{ \|D\mathbf F\|_{L^\infty(\Omega)}}. \end{equation}
I do not have any intuition about the above estimate. Moreover, I cannot find anything similar in the literature. Any help?
Just a little remark: the term $|\det D \mathbf F| D \mathbf F^{-T}= \operatorname{Cof} (D\mathbf F)$ is the cofactor matrix of $D\mathbf F$.