Divergence form

dolfinx
1

I want to compute the divergence of some vector functions.
a=\nabla \cdot(gf \nabla \rho)
If I create a functionspace V, and \rho,f,g are the interpolated functions of V, how to compute a? I hope that a is also an interpolated functions in V.
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V = fem.functionspace(domain, (“Lagrange”, 1))

rho = fem.Function(V)

rho.interpolate(lambda x: x[0]**2 + x[1]**2) # Example expression for rho

grad_rho = ufl.grad(rho)

div_expr = ufl.div(grad_rho)

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2

It seems that this does not work well.

3

Of course it doesn’t: you are computing the second derivative of a P1 function, hence the result will be zero.

5

Thanks for you reply. So is it possible to compute and restore the gradient of \rho in the 1 degree space? In that case I do not need to build a high degree function space.

6

You can get the gradient of \rho, which will be a piecewise constant function, thus it divergence would be zero.
If you use a second order lagrange function for \rho; the gradient will be piecewise linear (discontinuous across element boundaries), and the divergence (within each element) will be a piecewise constant.

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