How to apply pointsource to LinearVariationalProblem

Question

Hi all,

How does one apply a pointsource to a LinearVariationalProblem?

problem = LinearVariationalProblem(a, L, u, bc)

Normally I'd assemble L and then pointsource.apply works. But LinearVariationalProblem cannot take assembled version of L as input, so how can it be done int this case?

Thanks

Answer 1

As far as I'm aware, there is no functionality to include a Dirac delta function in a variational formulation in ufl/FEniCS.

Consider two approaches:

1. Form the FE matrix equation and apply the point source as you have
   already.

2. Use an approximation of the Dirac delta function in the variational
   formulation with a suitably fine mesh

$$ \delta(x) = \text{lim}_{\epsilon \rightarrow 0} \frac{1}{\epsilon \sqrt{\pi}} e^{-\left(\frac{x}{\epsilon}\right)^2} $$

Comments

Thank you very much for your suggestion! Do you know why it has been implemented when using ps.apply(assemble(L)) but not for ufl?
Furthermore, when trying the two approaches the obtained solutions are not the same, how can this be resolved?

I already tried the solution

f_q= Expression("sqrt((x[0] - x0)(x[0] - x0) + (x[1] - y0)(x[1] - y0) ) < r ? q : 0.0", x0=x0, y0=y0, r=thresh, q = 1, degree=2)

but yours is nicer, thanks!