Normally in any numerical solution of PDEs, you have the conjugate pairs, say, (displacement, force), OR (pressure, velocity) that contribute to the energy of the system. Let's notate these abstractly as (degree of freedom, its conjugate), or (DOF, CJG). In order to cover a wide range of application, one should check for convergence of
Relative residual (of the conjugate) values against some reference value, say L2-norm of the residual in system,
Iterate value of the DOF against some reference value, could be L2 or L1-norm of DOF in the system, and
Verify that $\Delta $Energy due to "incremental DOF times residual" is small compared to, say, the energy input to the system.
1 verifies that the residual goes to zero relative to some norm, 2 helps when large changes in DOFs are seen even at low values of residuals, and 3 ensures there is energy balance in the system.
It is very rare to check for absolute value of residuals (at least in solid mechanics, my field).