Finite element approximation and preconditioning for anisothermal flow of implicitly-constituted non-Newtonian fluids

Farrell, Patrick; Gazca Orozco, Pablo Alexei; Süli, Endre

doi:10.1090/mcom/3703

Finite element approximation and preconditioning for anisothermal flow of implicitly-constituted non-Newtonian fluids
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by Patrick Farrell, Pablo Alexei Gazca Orozco and Endre Süli HTML | PDF

Math. Comp. 91 (2022), 659-697 Request permission

Abstract:

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.

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