Finite element methods for coupled problems in incompressible fluid mechanics

Nonlinear coupled problems in fluid mechanics are crucial for modelling complex systems, where the fluid interacts with other components. The numerical solution of these coupled problems present many challenges. For example, the numerical method should preserve important physical features and respect its incompressibility. This usually requires a fine mesh (i.e., it is necessary to approximate the solution at many points in space), which in turn leads to very large linear and nonlinear systems of equations that require significant computational resources to solve. Consequently, capturing the detail required in realistic situations is prohibitively expensive using traditional numerical methods.
Advances in incompressible finite elements have shown that very accurate solutions to coupled fluid flow problems can be obtained using relatively coarse meshes. These new finite element methods respect the incompressibility condition exactly (or at least up to machine precision). This has the (undesirable) effect that the linear systems associated to them become very challenging to solve. This project's main aim is to overcome this drawback by developing accurate and reliable linear solvers, and to extend the applicability of incompressible finite element methods to more challenging situations.
Background:
Most realistic situations are modelled by coupled nonlinear partial differential equations (PDEs). For example, in the forming of new materials, the PDEs are those of thermoplasticity (nonlinear plasticity coupled with temperature). In this project we are interested in fluids whose viscosity varies with temperature. Applications arise in oceans and climate, advanced manufacturing (for example, during quenching), drug delivery through the blood vessels, etc. The equations describing this variable-viscosity flow are the incompressible Navier-Stokes equations coupled with temperature.
The complexity of these coupled problems generally necessitates solution by numerical methods, typically the finite element method. However, for the problem of interest two main difficulties arise. First, the problem is strongly coupled, since heat is convected by the fluid, while the temperature influences the flow. Also, to guarantee numerical stability (and physical conservativity) the fluid velocity must be incompressible (to guarantee local mass conservation). Additionally, the discretisation leads to a very large system of nonlinear equations that must be solved iteratively, making efficient solvers a must.
These two difficulties are independent, but related. More precisely, in recent work by one of the supervisors (GRB) a new strategy to compute incompressible discrete velocity fields, via ad-hoc post processing of classical finite element methods, was proposed. The advantages are numerous, but the most significant is the recovery of physical features of the continuous solution that usually need extremely refined meshes to be approximated well. The new approach recovers these features in meshes that are considerably coarser, and hence more computationally tractable. However, the incompressibility constraint must be satisfied exactly which means that certain equations of the linear system must be solved very accurately (up to machine precision). This is sometimes achievable by direct methods, but when the problem becomes large iterative methods are needed, which do not usually solve the equations to high accuracy. Thus, iterative methods have been linked to inaccuracies in some numerical experiments describing realistic situations, especially at very large Rayleigh numbers.