Non-converging concentration

[kamensky]

To expand on @dokken’s point, see my answer here:

Poisson equation the stubborn way

I second Nate’s suggestion to take some time with the fundamentals. To answer the questions specifically, Why this error? Because dx is really a sum of integrals over interiors of elements. You can see that, on the interior of each element, -\nabla\cdot(\nabla\phi) is zero when \phi is piecewise linear. Thus, the left-hand side matrix is singular. Why is inner(grad(phi), grad(v)) better? Because the other definition is not correct! Glossing over some technicalities about the theory o…

Also, note that including the boundary term from integration-by-parts (assuming there isn’t a DirichletBC, in which case d would be zero) is essentially leaving the boundary condition unspecified. Without choosing a specific boundary condition, there are many possible solutions satisfying the interior equation, so the problem for c will be ill-posed.