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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 ...


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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.


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