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Face midpoint value approximation
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face AB
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\( -\frac{1}{2}R\Gamma \left(\cos y-x\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)
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–
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face BC
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\( O\left({h}_{\theta}^2\right) \)
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face CD
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\( -\frac{1}{2}R\Gamma \left(\cos y-x\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)
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face DA
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\( O\left({h}_{\theta}^2\right) \)
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Gradient reconstruction
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error of \( \frac{\partial f}{\partial x} \)
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\( -R\Gamma \left( xy\cos xy+\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)
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\( -R\Gamma \left( xy\cos xy+\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)
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error of \( \frac{\partial f}{\partial y} \)
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\( -\frac{\Gamma}{2}\left(\cos y-x\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)
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\( -\frac{R\Gamma}{2\left(1+{\Gamma}^2\right)}\left(\left({y}^2+{x}^2{\Gamma}^2\right)\cos xy+\sin x+{\Gamma}^2\sin y\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)
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