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Face midpoint value approximation
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face AB
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\( \frac{1}{72}\left(-{x}^2\cos xy-\sin y\right){dx}^2+O\left({dx}^3\right) \)
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–
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face BC
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\( \frac{1}{12}\left(\cos x-\cos y+\left(x-y\right)\sin xy\right) dx+O\left({dx}^2\right) \)
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face AC
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\( \frac{1}{12}\left(-\cos x-\cos y+\left(x+y\right)\sin xy\right) dx+O\left({dx}^2\right) \)
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Gradient reconstruction
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error of \( \frac{\partial f}{\partial x} \)
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\( \frac{1}{3}\left(\cos x-y\sin xy\right)+O(dx) \)
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\( \frac{1}{3}\left(- xy\cos xy-\sin xy\right) dx+O\left({dx}^2\right) \)
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error of \( \frac{\partial f}{\partial y} \)
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\( \frac{1}{3}\left(-\cos y+x\sin xy\right)+O(dx) \)
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\( \frac{1}{18}\left(-\left({x}^2+2{y}^2\right)\cos xy-2\sin x-\sin y\right) dx+O\left({dx}^2\right) \)
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