Finite Element Methods (FEMs)
In the following, we will consider the Poisson problem with Dirichlet condition (homogeneous or non-homogeneous):
Problem : Find \(u : \Omega \rightarrow \mathbb{R}^d\) such that
\[\left\{
\begin{aligned}
-\Delta u &= f, \; &&\text{in } \; \Omega, \\
u&=g, \; &&\text{on } \; \partial\Omega,
\end{aligned}
\right.\]
with \(\Delta\) the Laplace operator and \(\Omega\subset\mathbb{R}^d\) a smooth bounded open set (and \(\partial\Omega\) its boundary).
Here, we present the 2 finite element methods we will be considering. First, we will present the standard FEM method in Section "Standard FEM", followed by the \(\phi\)-FEM method in Section "\(\phi\)-FEM".
The features include