My anecdotal experience with this is that it’s fine to use the full Jacobian of the SUPG terms in Newton’s method, although many manual implementations simply treat \tau as a constant when computing the Jacobian, which involves slightly fewer floating-point operations during assembly and is easier to implement when you don’t have access to FEniCS’s derivative function. However, using the exact Jacobians of highly-nonlinear shock-capturing viscosities is likely to lead to convergence issues.
1 Like