  We devise 3-field and 4-field finite element approximations of a system
describing the steady state of an incompressible heat-conducting fluid with
implicit non-Newtonian rheology. We prove that the sequence of numerical
approximations converges to a weak solution of the problem. We develop a block
preconditioner based on augmented Lagrangian stabilisation for a discretisation
based on the Scott-Vogelius finite element pair for the velocity and pressure.
The preconditioner involves a specialised multigrid algorithm that makes use of
a space-decomposition that captures the kernel of the divergence and
non-standard intergrid transfer operators. The preconditioner exhibits robust
convergence behaviour when applied to the Navier-Stokes and power-law systems,
including temperature-dependent viscosity, heat conductivity and viscous
dissipation.
