### What is the best way to calculate the divergent of a particular unit vector?

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Dear all,

How can I integrate the relationship above? (\( \phi\) is known)

\begin{equation}

\nabla\cdot\frac{\nabla \phi}{|\nabla \phi|} = \nabla\cdot\vec{n}

\end{equation}

This is the algorithm:

How can I integrate the relationship above? (\( \phi\) is known)

\begin{equation}

\nabla\cdot\frac{\nabla \phi}{|\nabla \phi|} = \nabla\cdot\vec{n}

\end{equation}

This is the algorithm:

```
F = FiniteElement("Lagrange", mesh.ufl_cell(), 2)
V = VectorFunctionSpace(mesh, "DG", 0, dim=2)
FE = FunctionSpace(mesh,F)
# Norma of the gradient
norm_grad = project(sqrt(phi0.dx(0)**2.0 + phi0.dx(1)**2.0), FE)
# Normal vector
n = project(grad(phi0)/(sqrt(phi0.dx(0)**2.0 + phi0.dx(1)**2.0)), V)
# Divergent of n
dver = div(n)
# our divergent of the gradient
dver = div(grad(phi0))/norm_grad
```

Community: FEniCS Project

### 1 Answer

3

You can do this, which also looks cleaner:

```
normal_phi = grad(phi)/sqrt(dot(grad(phi), grad(phi)))
out = assemble(div(normal_phi)*dx)
```

Integration is performed by assemble, and will only work if the result is scalar, i.e, phi must be a scalar function.

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