Dear Stephen,

At hanging nodes, there is definitely going to be a larger error due to the approximation of the diagonal mass matrix. I do not remember the exact details but to get a diagonal mass matrix you need to assume an interpolation in addition to the approximation leading to the mass lumping. If I remember correctly my co-worker Katharina Kormann describes this in her paper:
https://doi.org/10.4208/cicp.101214.021015a

So in general, I would assume that a change of 1e-8 as you have in your picture is simply the approximation error that occurs when going from the initially interpolated solution to the new mesh. I would assume that it goes away if you compute an "approximate" L2 projection where you compute a right hand side (v, u_init) for test functions v and the initial field u_init from ExactSolution of step-48 rather than VectorTools::interpolate.

Have you tried that?

If the change is larger in other cases as you describe, I would be more worried. Have you tried to investigate more? Does the projection instead of interpolation help in that case, too?

The implementation should be correct as far as I can tell - we have verified it on a large series of applications. In my experience, the most tricky thing to get right regarding constraints and boundary conditions is for nonlinear problems or implicit solvers with inhomogeneous boundary data, but that appears to not be the case here...

Best,
Martin


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