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

Journal article

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.

Authors: Farrell, P; Orozco, PAG; Suli, E

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: American Mathematical Society
• Journal: Mathematics of Computation
• Volume: 91
• Pages: 659-697
• Publication date: 2021-11-23
• Acceptance date: 2021-09-14
• DOI: 10.1090/mcom/3703
• EISSN: 1088-6842
• ISSN: 0025-5718
• Language: English
• Keywords: cs.NA; FFR; math.NA; 65N30, 65F08, 65N55, 76A05