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We devise 3-field and 4-field finite element approximations of a systemdescribing the steady state of an incompressible heat-conducting fluid withimplicit non-Newtonian rheology. We prove that the sequence of numericalapproximations converges to a weak solution of the problem. We develop a blockpreconditioner based on augmented Lagrangian stabilisation for a discretisationbased on the Scott-Vogelius finite element pair for the velocity and pressure.The preconditioner involves a specialised multigrid algorithm that makes use ofa space-decomposition that captures the kernel of the divergence andnon-standard intergrid transfer operators. The preconditioner exhibits robustconvergence behaviour when applied to the Navier-Stokes and power-law systems,including temperature-dependent viscosity, heat conductivity and viscousdissipation.

Finite element appoximation and augmented Lagrangian preconditioning for anisothermal implicitly-constituted non-Newtonian flow