1
If you want to use the coefficient of the form
$$
\alpha = \begin{cases}
\alpha_1 & r < 0.5 \\
\alpha_2 & r > 0.5
\end{cases}
$$
it is useful to work in polar coordinates. You build two pieces of the solution
$$
u_1(r,\theta), \qquad r < 0.5
$$
and
$$
u_2(r,\theta), \qquad r > 0.5
$$
Then at the interface you ensure solution and flux ...
1
If you want to know the why and the how, you probably want to watch lecture 21.65 here:
https://www.math.colostate.edu/~bangerth/videos.html
Which you probably want to do after you've watched lecture 21.6.
Disclaimer: I'm the one in these videos.
Only top voted, non community-wiki answers of a minimum length are eligible
