I have also noticed this issue where heat fluxes are computed many orders of magnitude too large when using SUPG. I have also been having problems with SUPG in 3D (both cartesian and spherical geometries, both with and without AMR). In every model that I have run, there are large (30%) temperature overshoots and undershoots and eventually the temperature solver will fail to converge, terminating the simulation. Increasing the ‘GMRES solver restart length’ will delay but not prevent crashing. I do have a relatively simple 3D test case that will produce an error but it takes about a day to run on 256 processors. Timo - would this be helpful?