I want to derive weak form of the Poisson's equation. I saw this article, but didn't help much.
$$
-\frac{\partial}{\partial x} \bigg( \frac{\partial u}{\partial x} \bigg )
-\frac{\partial}{\partial y} \bigg( \frac{\partial u}{\partial y} \bigg )
= f \hspace {10pt} \text {in } \Omega
$$
I started by multiplying by weight function $w$ and integrating it over $XY$ space. So,
$$
w\int_\Omega \bigg[ -\frac{\partial}{\partial x} \bigg( \frac{\partial u}{\partial x} \bigg )
-\frac{\partial}{\partial y} \bigg( \frac{\partial u}{\partial y} \bigg )
– f \bigg] dxdy =0
$$
Next step should be using integration by parts using identity
$ \int udv = uv – \int vdu $. But I am not getting how to use this in the above equation. If I just consider this part of the equation,
$$
w\int_\Omega -\frac{\partial}{\partial x} \bigg( \frac{\partial u} {\partial x} \bigg ) dxdy=0
$$
I don't understand how to proceed further as integration is in $x,y$. Any help is appreciated.
Best Answer
It is possible to derive the 'multidimensional integration by parts formula' or the Green's formula using the Gauss divergence theorem. Let $a$ and $b$ be arbitrary smooth enough functions. We have by the product rule (using index notation) $$\int_{\Omega} (ab)_{,j} \,\mathrm{d}x = \int_{\Omega} a_{,j} b + ab_{,j} \,\mathrm{d}x.$$ Using Gauss divergence on the LHS term gives $$\int_{\Omega} ab n_j \,\mathrm{d}x = \int_{\Omega} a_{,j} b + ab_{,j} \,\mathrm{d}x.$$ Rearranging the terms gives $$\int_{\Omega} a_{,j} b \,\mathrm{d}x = \int_{\Omega} ab n_j \,\mathrm{d}x - \int_{\Omega} ab_{,j} \,\mathrm{d}x.$$ Notice the similarity of this and the one-dimensional integration by parts formula (hence the casual nomenclature).
Apply this in your second equation with $b \rightarrow w$ and $a \rightarrow \frac{\partial u}{\partial x}$ or $a \rightarrow \frac{\partial u}{\partial y}$ depending on the term. The boundary term will either vanish or not vanish depending on the boundary condition (for $u=0$ on boundary it will vanish). By the way, your $w$ should be inside the integral, not outside.