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Table 7 Discretization errors of gradient reconstruction and face midpoint value approximation (reg. tri)

From: Accuracy analysis of gradient reconstruction on isotropic unstructured meshes and its effects on inviscid flow simulation

Discretization errors GG-Cell LSQ-basic
Face midpoint value approximation face AB \( \frac{1}{72}\left(-{\left(-2x+y\right)}^2\cos xy-\sin x-4\sin y+4\sin xy\right){dx}^2+O\left({dx}^3\right) \)
face BC \( \frac{1}{72}\left(-{\left(x-2y\right)}^2\cos xy-4\sin x-\sin y+4\sin xy\right){dx}^2+O\left({dx}^3\right) \)
face AC \( \frac{1}{72}\left(-{\left(x+y\right)}^2\cos xy-\sin x-\sin y-2\sin xy\right){dx}^2+O\left({dx}^3\right) \)
Gradient reconstruction error of \( \frac{\partial f}{\partial x} \) \( \frac{1}{6}\left(\left(2x-y\right)y\cos xy-\sin x+2\sin xy\right) dx+O\left({dx}^2\right) \) \( \frac{1}{6}\left(\left(2x-y\right)y\cos xy-\sin x+2\sin xy\right) dx+O\left({dx}^2\right) \)
error of \( \frac{\partial f}{\partial y} \) \( \frac{1}{6}\left(-x\left(x-2y\right)\cos xy-\mathrm{siny}+2\sin xy\right) dx+O\left({dx}^2\right) \) \( \frac{1}{6}\left(-x\left(x-2y\right)\cos xy-\mathrm{siny}+2\sin xy\right) dx+O\left({dx}^2\right) \)