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Singuarity error and mesh size

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Hi
I have a pde based model where


It is a 1d problem which simplifies it to an extent. when I try to run it at coarser mesh sizes I get an error saying may have reached a singularity. When the mesh I use is made finer the singularity error occurs at a lower value of time.

I have been reading on trying to reduce the singularity error and I think that it might be because of gamma which is a derivative of u (du/dr). There are sharp point in the u vs r graph, so I think this might be the problem.

Can anybody help me in solving this problem. Is it because of the meshing. So how do I get rid of the singularity error by proper meshing of the 1d problem. I know that the singularity always occurs at r-around 0.5. How can I increase the meshing there.

Thank you for taking the time to help me out.

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