Study guide: Introduction to finite element methods

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Integration by parts

Rule for multi-dimensional integration by parts.

$$ \begin{equation*} -\int_{\Omega} \nabla\cdot (a(\x)\nabla u) v\dx = \int_{\Omega} a(\x)\nabla u\cdot\nabla v \dx - \int_{\partial\Omega} a\frac{\partial u}{\partial n} v \ds \end{equation*} $$

$ \int_\Omega ()\dx $: area (2D) or volume (3D) integral

$ \int_{\partial\Omega} ()\ds $: line(2D) or surface (3D) integral

$ \partial\Omega_N $: Neumann conditions $ -a\frac{\partial u}{\partial n} = g $

$ \partial\Omega_D $: Dirichlet conditions $ u = u_0 $

$ v\in V $ must vanish on $ \partial\Omega_D $ (in method 1)