I am reading the book Numerical Solution of Partial Differential Equations by the Finite Element Method by Claes Johnson.
In Chapter 1 he talks about the Possion Equation, and to prove that FEM solves this, he starts by defining divergence:
$$\int_{\Omega} div(A)\; dx = \int_{\Omega} \frac{\partial A_1}{\partial x_1} + \frac{\partial A_1}{\partial x_2} \; dx = \int_{\Gamma} A \cdot n \; ds$$
And says that if we apply this to A = (vw,0) and A = (0,vw) we get that:
$$\int_{\Omega} div(vw,0) \; dx = \int_{\Omega} \frac{\partial vw}{\partial x_1} + 0 \; dx = \int_{\Omega} \frac{\partial v}{\partial x_1}w \; dx + \int_{\Omega} \frac{\partial w}{\partial x_1} v \; dx = \int_{\Gamma} vw \cdot n_1 \; ds$$
$$\int_{\Omega} div(0,vw) \; dx = 0 + \int_{\Omega} \frac{\partial vw}{\partial x_2} \; dx = \int_{\Omega} \frac{\partial v}{\partial x_2}w \; dx + \int_{\Omega} \frac{\partial w}{\partial x_2} v \; dx = \int_{\Gamma} vw \cdot n_2 \; ds$$
These equations I follow and understand, but the next part is where I give up!
He defines the gradient as $\nabla v = \left( \frac{\partial v}{\partial x_1} + \frac{\partial v}{\partial x_2} \right) $
And by using the two equations from above we can "see" that:
$$\int_{\Omega} \nabla v \cdot \nabla w \; dx \equiv \int_{\Omega} \left[ \frac{\partial v}{\partial x_1} \frac{\partial w}{\partial x_1} + \frac{\partial v}{\partial x_2} \frac{\partial w}{\partial x_2} \right] dx = \int_{\Gamma} \left[ v \frac{\partial w}{\partial x_1}n_1 + v \frac{\partial w}{\partial x_2}n_2 \right] ds – \int_{\Omega} v\left[ \frac{\partial^2 w}{\partial x_1^2} + \frac{\partial^2 w}{\partial x_2^2} \right] dx$$
And it's this last equation that I simply can't figure out. I can see that the two gradients (vectors multiplied) gives the "congruent" with, but from here I'm lost!
Is there anyone who can give me a hint to how to use the theorem/equations from before??
Any help is much appreciated.
Best Answer
Note that $\nabla v$ is a vector, not a scalar: $$ \nabla v = \left(\frac{\partial v}{\partial x_1}, \frac{\partial v}{\partial x_2}\right) $$
Now observe that: \begin{align*} div(v \nabla w) &= div\left(v\frac{\partial w}{\partial x_1}, v\frac{\partial w}{\partial x_2}\right) \\ &= \frac{\partial}{\partial x_1}\left[ v\frac{\partial w}{\partial x_1} \right] + \frac{\partial}{\partial x_2}\left[ v\frac{\partial w}{\partial x_2} \right] \\ &= \left[\frac{\partial v}{\partial x_1}\frac{\partial w}{\partial x_1} + v\frac{\partial^2 w}{\partial x_1^2} \right] + \left[\frac{\partial v}{\partial x_2}\frac{\partial w}{\partial x_2} + v\frac{\partial^2 w}{\partial x_2^2} \right] \\ &= \left[\frac{\partial v}{\partial x_1}\frac{\partial w}{\partial x_1} + \frac{\partial v}{\partial x_2}\frac{\partial w}{\partial x_2} \right] + v\left[\frac{\partial^2 w}{\partial x_1^2} + \frac{\partial^2 w}{\partial x_2^2} \right] \\ &= [\nabla v \cdot \nabla w] + v\left[\frac{\partial^2 w}{\partial x_1^2} + \frac{\partial^2 w}{\partial x_2^2} \right] \\ \end{align*}
Hence, it follows that: \begin{align*} \int_{\Omega} \nabla v \cdot \nabla w \, dx &= \int_{\Omega} \left( div(v \nabla w) - v\left[\frac{\partial^2 w}{\partial x_1^2} + \frac{\partial^2 w}{\partial x_2^2} \right]\right) \, dx \\ &= \int_{\Omega} div(v \nabla w) \, dx - \int_{\Omega} v\left[\frac{\partial^2 w}{\partial x_1^2} + \frac{\partial^2 w}{\partial x_2^2} \right] \, dx \\ &= \int_{\Gamma} (v \nabla w) \cdot n \, ds - \int_{\Omega} v\left[\frac{\partial^2 w}{\partial x_1^2} + \frac{\partial^2 w}{\partial x_2^2} \right] \, dx \\ &= \int_{\Gamma} \left[ v \frac{\partial w}{\partial x_1}n_1 + v \frac{\partial w}{\partial x_2}n_2 \right] \, ds - \int_{\Omega} v\left[ \frac{\partial^2 w}{\partial x_1^2} + \frac{\partial^2 w}{\partial x_2^2} \right] \, dx \end{align*}