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

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.

Structure-Preserving and Helicity-Conserving Finite Element Approximations and Preconditioning for the Hall Mhd Equations

“…While the literature on preconditioners for anisothermal MHD models does not seem to be too rich, the preconditioning of standard Rayleigh-Bénard problems or thermal convection-driven flows is more well-known, cf. [38,59] or [37] for an overview.…”

“…For the top left block, we use a similar idea that was introduced in [38]. The first step is to perform a further Schur complement approximation grouping together where instabilities in the velocity were mainly aligned with the magnetic field.…”

Weak-strong Uniqueness for Heat Conducting non-Newtonian Incompressible Fluids

“…[12]). When tackling the question of existence of dissipative weak solutions for implicit models one could employ a finite element formulation including the stress S S S as an unknown, in an analogous way to the formulations analysed in [32,28].…”

Non-isothermal non-Newtonian fluids: the stationary case

“…Concerning non-isothermal Newtonian fluid problems, we refer to [6], where spectral elements have been employed, and to [30] which considers the finite element approximation of the heat equation coupled with Stokes equations under threshold type boundary condition. It is worth mentioning that a very recent contribution is devoted to the numerical approximation of the steady state problem for an isothermal incompressible heat-conducting fluid with implicit non-Newtonian rheology (see [33]). There, it is proven that the sequence of numerical approximations converge to a weak solution to the original problem.…”