Weak form Poisson

The partial differential equation can be written as:

\[\begin{aligned} -\nabla^2 u = f\end{aligned}\]

Multiply by test function \(v\).

\[\begin{aligned} -\nabla^2 u v = f v\end{aligned}\]

Integrate on both sizes.

\[\begin{aligned} - \int_{\Omega} \nabla^2 u v \, d\Omega = \int_{\Omega} f v \, d\Omega\end{aligned}\]

Integrate by parts.

\[\begin{aligned} \int_{\Omega} \nabla^2 u v \, d\Omega = \int_{\Omega} \nabla \cdot \nabla u v \, d\Omega = - \int_{\Omega} \nabla u \nabla v \, d\Omega + \int_{\Omega} \nabla \cdot \left( \nabla u \, v \right) \, d\Omega\end{aligned}\]

Apply Gauss divergence theorem.

\[\begin{aligned} \int_{\Omega} \nabla \cdot \left( \nabla u \, v \right) \, d\Omega = \int_{\Gamma_{N} \cup \Gamma_{D}} \nabla u v \cdot n\, d\Gamma\end{aligned}\]

Because \(v = 0\) on \(\Gamma_{D}\).

\[\begin{aligned} \int_{\Omega} \nabla \cdot \left( \nabla u \, v \right) n \, d\Omega = \int_{\Gamma_{N}} \nabla u v \cdot n\, d\Gamma\end{aligned}\]

The integration by parts becomes:

\[\begin{aligned} \int_{\Omega} \nabla^2 u v \, d\Omega = - \int_{\Omega} \nabla u \nabla v \, d\Omega + \int_{\Gamma_{N}} \nabla u v \cdot n\, d\Gamma\end{aligned}\]

Then:

\[\begin{aligned} \int_{\Omega} \nabla u \nabla v \, d\Omega - \int_{\Gamma_{N}} \nabla u v \cdot n\, d\Gamma = \int_{\Omega} f v \, d\Omega\end{aligned}\]

Theory

Integration by parts

\[\begin{aligned} \int_{\Omega} \left(\nabla \cdot a \right) b\,d\Omega = -\int_{\Omega} a \nabla b\,d\Omega + \int_{\Omega} \nabla \cdot \left( a b \right) n\,d\Omega \\\end{aligned}\]

Gauss divergence theorem

\[\begin{aligned} \int_{\Omega} \nabla \cdot F\, d\Omega = \int_{\partial \Omega = \Gamma_{N} \cup \Gamma_{D}} F \cdot n\, d\Gamma\end{aligned}\]