P1 approximation of a parabolic shear map

Direct differentiation gives $D\mathrm{\Phi }=\left(\begin{array}{cc}1& 0\\ x\mathrm{/}2& 1\end{array}\right)$ and determinant $1$; the stated inverse is verified by substitution. On a triangle with barycentric coordinates ${\lambda }_i$ and vertex first-coordinates $x_i$, exact quadratic expansion gives $I_h(x^2\mathrm{/}4)-x^2\mathrm{/}4=\frac{1}{4}{\sum }_{i<j}{\lambda }_i{\lambda }_j(x_i-x_j{)}^2$. Since $\mathrm{\mid }{x}_i-x_j\mathrm{\mid }\le h_K$ and $0\le {\lambda }_i\le 1$, this is bounded by $C{h}_K^2$.

Differentiating the barycentric expression in step 1.1, only the factors ${\lambda }_i{\lambda }_j$ vary; their derivatives are bounded by $C{h}_K^{-1}$ using [F2], while $\mathrm{\mid }{x}_i-x_j{\mathrm{\mid }}^2\le h_K^2$. The product rule therefore gives a derivative bound $C{h}_K$. Thus $\mathrm{\parallel }D(I_h(x^2\mathrm{/}4)-x^2\mathrm{/}4){\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}\le C{h}_K$, yielding the global $Ch$ derivative estimate and the $C{h}^2$ value estimate. By [F3] these are the geometry errors $e_1$ and $e_0$.

The first coordinate of ${\mathrm{\Phi }}_h$ is $x$, and on each vertical segment its second coordinate is the continuous piecewise-affine function $y+I_h(x^2\mathrm{/}4)(x,y)$. Its slope on every subsegment is $1+{\mathrm{\partial }}_y{I}_h(x^2\mathrm{/}4)\ge 1-Ch$. Choose $h$ with $Ch\le 1\mathrm{/}2$. Integration over the finitely many subsegments makes this coordinate strictly increasing with increment at least half the vertical increment. Equal images must therefore have equal first coordinates and then equal second-coordinate inputs; hence ${\mathrm{\Phi }}_h$ is injective. The same derivative estimate gives $\det D{\mathrm{\Phi }}_h=1+{\mathrm{\partial }}_y{I}_h(x^2\mathrm{/}4)\ge 1\mathrm{/}2$.