Mixed FEM for the Poisson problem#
Let \(\Omega \subset \mathbb{R}^3\) and consider the Poisson problem
\[\begin{split}
\begin{align}
\left\{
\begin{array}{clr}
-\Delta p & =f & ,~\Omega \\
p & =0 & ,~\partial \Omega
\end{array} \right.
\end{align}
\end{split}\]
Using that \(\Delta p = \nabla\cdot\nabla p\), we set \( \mathbf{u}=\nabla p\), then the Poisson equation can be written equivalently
\[ \mathbf{u}=-\nabla p, ~~~ \nabla\cdot \mathbf{u}= f.\]