Stokes方程式（Ｄフォーム）

おさらい
強形式＋境界条件

$\left{ \begin{cases} \dfrac{\partial \vec{u}}{\partial t} - 2 \nabla \cdot D(\vec{u}) + \nabla p &= \vec{f}\\ \nabla \cdot \vec{u} = 0 \end{cases} \right.$

弱形式

$(\dfrac{\partial \vec{u}}{\partial t},\vec{v}) + 2 (D(\vec{u}),D(\vec{v})) - (p,\nabla \cdot \vec{v}) = (\vec{f},\vec{v})$

離散化された弱形式

$\left( \frac{\vec{u}^{n+1}_h-\vec{u}^n_h}{\Delta t} , \vec{v}_h \right) + 2 \left( D \left( \vec{u}^{n+1}_h \right),D(\vec{v}_h) \right) - \left( p^{n+1}_h,\nabla \cdot \vec{v}_h \right) = (\vec{f}^{n+1}_h,\vec{v}_h)$

既知部分を移項して

$\left( \frac{\vec{u}^{n+1}_h}{\Delta t} , \vec{v}_h \right) + 2 \left( D \left( \vec{u}^{n+1}_h \right),D(\vec{v}_h) \right) - \left( p^{n+1}_h,\nabla \cdot \vec{v}_h \right) = (\vec{f}^{n+1}_h,\vec{v}_h) + \left( \frac{\vec{u}^n_h}{\Delta t} , \vec{v}_h \right)$

ここから要素剛性行列

拡散項

$2 (D(\vec{u}):D(\vec{v})) = 2 \frac{\partial u_1}{\partial x_1}\frac{\partial v_1}{\partial x_1} + \left( \frac{\partial u_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1} \right) \left( \frac{\partial v_1}{\partial x_2} + \frac{\partial v_2}{\partial x_1} \right) + 2 \frac{\partial u_2}{\partial x_2}\frac{\partial v_2}{\partial x_2}$

$= 2 \frac{\partial u_1}{\partial x_1}\frac{\partial v_1}{\partial x_1} + \frac{\partial u_1}{\partial x_2} \frac{\partial v_1}{\partial x_2} + \frac{\partial u_1}{\partial x_2} \frac{\partial v_2}{\partial x_1} + \frac{\partial u_2}{\partial x_1} \frac{\partial v_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1} \frac{\partial v_2}{\partial x_1} + 2 \frac{\partial u_2}{\partial x_2}\frac{\partial v_2}{\partial x_2}$