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)