有限要素法・流体力学による数値計算

\[ \begin{aligned} \Delta &=(\nabla\cdot\nabla)\\[6pt] &=\left(\sum_{i=0}^{3}\vec{\delta}_{i}\frac{\partial}{\partial x_i}\cdot\sum_{j=0}^{3}\vec{\delta}_{j}\frac{\partial}{\partial x_j}\right)\\[6pt] &=\left(\sum_{i=0}^{3}\sum_{j=0}^{3}\vec{\delta}_{i}\frac{\partial}{\partial x_i}\cdot\vec{\delta}_{j}\frac{\partial}{\partial x_j}\right)\\[6pt] &=\sum_{i=0}^{3}\sum_{j=0}^{3}(\vec{\delta}_{i}\cdot\vec{\delta}_{j})\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_j}\\[6pt] &=\sum_{i=0}^{3}\sum_{j=0}^{3}\delta_{ij}\frac{\partial}{\partial x_i x_j}\\[6pt] &=\sum_{i=0}^{3}\frac{\partial}{\partial x_i^2}\\[6pt] &=\frac{\partial}{\partial x^2}+\frac{\partial}{\partial y^2}+\frac{\partial}{\partial z^2} \end{aligned} \]