Numerical Analysis of Unsteady Implicitly Constituted Incompressible Fluids: Three-Field Formulation

Adaptive iterative linearised finite element methods for implicitly constituted incompressible fluid flow problems and its application to Bingham fluids

In this work, we introduce an iterative linearised finite element method for the solution of Bingham fluid flow problems. The proposed algorithm has the favourable property that a subsequence of the sequence of iterates generated converges weakly to a solution of the problem. This will be illustrated by two numerical experiments.

隐式构成理论下不可压缩流体不稳流动的全离散有限元近似

Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids

本文研究了隐式构成理论下不可压缩流体不稳流动的全离散有限元近似方法，采用混合有限元空间近似与后向欧拉时间步进结合，证明了在特定条件下近似解序列的收敛性，利用分裂与正则化技术、抛物型利普希茨截断方法、局部 Minty 型单调性结果及多种弱紧致性结果实现了最大范围 q 的存在性弱解。

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

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 diverge…

An augmented Lagrangian preconditioner for implicitly-constituted non-Newtonian incompressible flow

We propose an augmented Lagrangian preconditioner for a three-field stress-velocity-pressure discretization of stationary non-Newtonian incompressible flow with an implicit constitutive relation of power-law type. The discretization employed makes use of the divergence-free Scott-Vogelius pair for the velocity and pressure. The preconditioner builds on the work [P. E. Farrell, L. Mitchell, and F. Wechsung, SIAM J. Sci. Comput., 41 (2019), pp. A3073-A3096], where a Reynolds-robust preconditioner for the three-dimensional Newtonian system was introduced. The preconditioner employs a specialized …

A semismooth Newton method for implicitly constituted non-Newtonian fluids and its application to the numerical approximation of Bingham flow

We propose a semismooth Newton method for non-Newtonian models of incompressible flow where the constitutive relation between the shear stress and the symmetric velocity gradient is given implicitly; this class of constitutive relations captures for instance the models of Bingham and Herschel-Bulkley. The proposed method avoids the use of variational inequalities and is based on a particularly simple regularisation for which the (weak) convergence of the approximate stresses is known to hold. The system is analysed at the function space level and results in mesh-independent behaviour of the no…