3.2: Differentiation of Bivariate Functions
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Let us briefly consider differentiation of bivariate functions. For simplicity, we restrict ourselves to estimation of the first derivative, i.e., the gradient.
Example 3.2.1 first-order gradient
Following the procedure for estimating the first derivative of univariate functions, we first construct a polynomial interpolant and then evaluate its derivative. In particular, we consider a linear interpolant on a triangle. Recall the second form of linear interpolant, \[(\mathcal{I} f)(\boldsymbol{x})=f\left(\bar{x}^{\mathbf{1}}\right)+b^{\prime}\left(x-\bar{x}^{1}\right)+c^{\prime}\left(y-\bar{y}^{1}\right) .\] The partial derivative in the \(x\)-direction is \[\frac{\partial(\mathcal{I} f)}{\partial x}=b^{\prime}=\frac{1}{A}\left[\left(f\left(\overline{\boldsymbol{x}}^{2}\right)-f\left(\overline{\boldsymbol{x}}^{1}\right)\right)\left(\bar{y}^{3}-\bar{y}^{1}\right)-\left(f\left(\overline{\boldsymbol{x}}^{3}\right)-f\left(\overline{\boldsymbol{x}}^{1}\right)\right)\left(\bar{y}^{2}-\bar{y}^{1}\right)\right],\] where we recall that \(A\) is twice the area of the triangle, i.e., \(A=\left(x_{2}-x_{1}\right)\left(y_{3}-y_{1}\right)-\left(x_{3}-x_{1}\right)\left(y_{2}-y_{1}\right)\). Similarly, the derivative in the \(y\)-direction is \[\frac{\partial(\mathcal{I} f)}{\partial y}=c^{\prime}=\frac{1}{A}\left[\left(f\left(\overline{\boldsymbol{x}}^{3}\right)-f\left(\overline{\boldsymbol{x}}^{1}\right)\right)\left(\bar{x}^{2}-\bar{x}^{1}\right)-\left(f\left(\overline{\boldsymbol{x}}^{2}\right)-f\left(\overline{\boldsymbol{x}}^{1}\right)\right)\left(\bar{x}^{3}-\bar{x}^{1}\right)\right] .\] In general, a directional derivative in the direction \(s=\left(s_{x}, s_{y}\right)\) is \[\frac{\partial(\mathcal{I} f)}{\partial s}=s_{x} \frac{\partial(\mathcal{I} f)}{\partial x}+s_{y} \frac{\partial(\mathcal{I} f)}{\partial y}=s_{x} b^{\prime}+s_{y} c^{\prime}\] Because the gradient approximation is constructed from a linear function, the gradient estimate is constant over the triangle. The approximation is first-order accurate.