Ghost nodes

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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.

The code below constructs ghost nodes and remembers the corresponding ghost node for each boundary node.

Range<int> gh(domain.size(), -1);  // Maps boundary nodes to their corresponding ghost nodes.
for (int i : neu) {  // Add ghost nodes for each Neumann boundary node
    gh[i] = domain.addInternalNode(domain.pos(i) + dx*domain.normal(i), 2);  
}

When assembling the system,

for (int i : interior) {
    op.lap(i) = 1.0;  // Equation
}
for (int i : dir) {
    op.value(i) = 0.0;  // Dirichlet
}
for (int i : neu) {
    op.neumann(i, domain.normal(i)) = 0.0;  // Neumann
    // Equation holds also on the boundary, write it in the row corresponding to the ghost node.
    op.lap(i, gh[i]) = 1.0;
}

The solution is shown below:

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