Ghost nodes

See the Ghost nodes (theory) page for what ghost nodes and how they are used. We will use them in this example to reliably solver a 3D mixed Dirichlet and Neumann problem on an irregular domain.

We will solve the problem

$\nabla^2 u = 1 \text{ in } \Omega, \quad \frac{\partial u}{\partial n} = 0 \text{ on } \partial \Omega_{+}, \quad u = 0 \text{ on } \partial \Omega_{-}$

where $\Omega = B(\boldsymbol{0}, 1) - B(\boldsymbol{1}, 1.5)$, $\partial \Omega_{+}$ is the part of the boundary with the nonnegative $x$ coordinate and $\partial \Omega_{-}$ the part of the boundary with negative $x$ coordinate.

TODO code, explanation, matrix.

The solution is shown below: