I tried to implement the static heat equation in cylindrical coordinates on
a 2d-plane, while being radially symmetric at x = 0. That leads me to the
following weak formulation of the problem (while neglecting the factor
2\pi):
\int\limits_0^R \nabla U\nabla\phi r dr dz = \int\limits_0^R f\phi r dr dz
Compared to the cartesian version
\int\limits_0^Y \nabla U\nabla\phi dx dy = \int\limits_0^Y f\phi dx dy
the onliest difference is the factor r in the equation.
The question is now: When writing this into the program code, I iterate
over the dimensions. Does every dimension need the additional factor r,
i.e. do I have to multiply every dimension with r, or only the parts which
are responsible for the direction r is going? I would suspect the latter,
but I do not know how to do that for calculating the right hand side.
Usually I would expect (for two dimensions) that (when printing both
solutions separately) both solutions are equal, but when using a tensor for
applying the prefactors with
Tensor<1, dim> r_pref;
r_pref[0] = 1;
r_pref[1] = r_val; //value for r at the current point q
for(size_t d = 0; d < dim; ++d)
cell_rhs(i) += fe_values[surface].value(i, q)[d] * r_pref[d] * f *
fe_values.JxW(q);
I get an unsymmetric right hand side, and the solutions A_x and A_y are
different, too. Thus I assume that this approach is wrong. Is that correct,
or did I not consider something else?
Thanks!
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