A parabolic shear has certified linear geometry approximations

Differentiate $\mathrm{\Phi }$ to obtain $D\mathrm{\Phi }=\left(\begin{array}{cc}1& 0\\ x\mathrm{/}2& 1\end{array}\right)$ and $\det D\mathrm{\Phi }=1$. The quadratic term makes the image of a nonvertical straight edge a parabolic arc, while vertical edges remain vertical.

Apply [F2] on the shape-regular mesh to get the stated value error, derivative error, and positive-Jacobian certificate.

On each shared reference edge, the two affine traces of $I_h(x^2\mathrm{/}4)$ agree at its endpoints and therefore agree everywhere on that edge. Thus ${\mathrm{\Phi }}_h$ has matching traces; the exact map is globally smooth and has matching traces as well. On each boundary edge, ${\mathrm{\Phi }}_h$ is the straight segment joining the exact images of its vertices. The vertical boundary traces agree exactly with $\mathrm{\Phi }$; on each horizontal boundary segment, ${\mathrm{\Phi }}_h(x,\pm 1)=(x,\pm 1+I_h(x^2\mathrm{/}4))$, so the boundary discrepancy is bounded by $C{h}^2$ from step 1.2. Compatibility here does not assert equality of the exact parabolic arcs and their straight approximations. By [F2], ${\mathrm{\Phi }}_h$ is continuous and injective on the compact square, hence by [F4] is a homeomorphism onto its image, as is $\mathrm{\Phi }$. Its cell restrictions are invertible affine maps extending to neighborhood diffeomorphisms; the exact restrictions extend by the global shear. These facts verify matching and boundary-trace compatibility and common image topology in [F1].

Step 1.1 bounds $D\mathrm{\Phi }$ uniformly, and step 1.2 gives $\mathrm{\parallel }D{\mathrm{\Phi }}_h{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }D\mathrm{\Phi }{\mathrm{\parallel }}_{\mathrm{\infty }}+Ch$ and determinant lower bounds $J=1$, $J_h\ge 1\mathrm{/}2$. Apply [F3] cellwise, with the compatible maps from step 2.1, and substitute $e_1\le Ch$ from step 1.2. This yields $\mathrm{\parallel }B-B_h{\mathrm{\parallel }}_{\mathrm{\infty }}\le Ch$ with constants uniform over the mesh family.