Cross derivatives in FEniCS

variational formulation
1

Hello. I’m trying to write the weak form of the following simplified equation, where w is a test funcion

\large \iint(w \frac{\partial ^2 u}{\partial x ^2})dxdy + \iint(w \frac{\partial ^2 u}{\partial x \partial y})dxdy = 0

I know from Green’s first identity that

\large \iint(w \frac{\partial ^2 u}{\partial x ^2})dxdy = \int (wu\hat{n}_x)ds - \iint (\frac{\partial w}{\partial x} \frac{\partial u}{\partial x})dxdy

But what about the other term? Should I let it untouched and access the cross derivatives through a command like (w*u.dx(0)*u.dx(1))dx (I’m not sure if it can be done)? Or is there any other theorem I’m not aware of to deal with this integral?

2

Firt you’ve lade a mistake… What matters with green is on which variable you do the integration.
\iint(w \frac{\partial^2 u}{\partial x\partial y}) \mathrm{d}x\mathrm{d}y= \int w \frac{\partial u}{\partial y}\hat{n}_x \mathrm{d}s-\iint \frac{\partial w}{\partial x} \frac{\partial u}{\partial y}
So for the first term:
\iint(w \frac{\partial^2 u}{\partial x^2}) \mathrm{d}x\mathrm{d}y= \int w \frac{\partial u}{\partial x}\hat{n}_x \mathrm{d}s-\iint \frac{\partial w}{\partial x} \frac{\partial u}{\partial x}