manifold, we apply a Dirichlet boundary condition by fixing $u_i$ on each boundary node. We make the convention that the cross field index of a boundary node (see [@viertel_approach_2019]) is assigned according to .

[p[2cm]{}p[2.4cm]{}p[2cm]{}p[4.9cm]{}]{} Interior Angle & Index\
$(0, \frac{3\pi}{4})$ & $\frac{1}{4}$\
$[\frac{3\pi}{4}, \frac{5\pi}{4}]$ & 0\
$(\frac{5\pi}{4}, \frac{7\pi}{4}]$ & $-\frac{1}{4}$\
$(\frac{7\pi}{4}, 2\pi)$ & $-\frac{1}{2}$\

For a boundary node $n_i$ of index 0 or $-\frac{1}{2}$, we compute the outward pointing unit normal vector of each boundary edge adjacent to $n_i$ lying in the plane of the adjacent boundary face. We then project these vectors into the tangent plane at $n_i$, and bisect them with a unit vector, $d_i$, which averages the directions of the facet normals. Since we want to align the cross to this vector, we set $u_i = d_i^4$. For nodes of index $\pm$ $\frac{1}{4}$, $d_i$ is first rotated $\pi/4$ radians about $\mathbf{n}_i$ in the positive direction before computing $u_i$.

### The Diffusion Generated Method for Cross Field Design

Intuitively, we would like to solve until stationarity the time dependent problem $$\begin{aligned}
 \label{eq:time-dependent}
  u_t(t,x) &= \Delta u(t,x) && x \in M\nonumber\\
  u(t,x) &= g(x) && x \in \partial M\\
  u(0,x) &= u^0(x) && x \in M\nonumber\end{aligned}$$

with the 