Extended finite element method

Remarks

The extra functions $N_j \psi$ have small support, so the system matrix stays sparse. If all $a_j$ equal one value $a$, then on a region where $\sum_{j \in J} N_j = 1$ the enrichment equals $a \psi$ exactly. Common choices of $\psi$ are a Heaviside function across a crack, and the singular functions of the crack tip, which behave like $\sqrt{r}$. Often one uses the shifted form $\psi(x) - \psi(x_j)$, so that $u_h(x_i) = u_i$ at the nodes.

Example: on $\Omega = (0, 1)$ with nodes $x_i = ih$, let a crack sit at a point $c$ inside $(x_k, x_{k+1})$. Take $\psi(x) = H(x - c)$, with $H$ the Heaviside step, and $J = \set{k, k+1}$. Then $u_h$ can jump at $c$ by $N_k(c) a_k + N_{k+1}(c) a_{k+1}$, while the mesh does not have a node at $c$.

Non-example: the standard space with only the first sum is continuous on $\Omega$. It has no enrichment, $J = \emptyset$, so it cannot jump at $c$ unless $c$ is a mesh node and the mesh is split there.