Finite element methods for coupled problems in incompressible fluid mechanics

流体力学中的非线性耦合问题对于模拟复杂系统至关重要，其中流体与其他组件相互作用。这些耦合问题的数值解提出了许多挑战。例如，数值方法应保留重要的物理特征，并尊重其不可压缩性。这通常需要细网格（即，必须在空间中的许多点处近似解），这又导致需要大量计算资源来求解的非常大的线性和非线性方程组。因此，在现实情况下捕捉所需的细节是昂贵的，使用传统的数值方法。不可压缩有限元的进展表明，使用相对粗糙的网格可以获得耦合流体流动问题的非常精确的解。这些新的有限元方法严格遵守不可压缩性条件（或至少达到机器精度）。这具有（不期望的）效果，即与它们相关联的线性系统变得非常难以求解。这个项目的主要目的是克服这个缺点，开发准确可靠的线性求解器，并延长不可压缩有限元方法的适用性，以更具挑战性的situation.Background：最现实的情况是由耦合非线性偏微分方程（PDE）建模。例如，在新材料的成形中，偏微分方程是热塑性（与温度耦合的非线性塑性）的偏微分方程。在这个项目中，我们感兴趣的是粘度随温度变化的流体。应用出现在海洋和气候，先进的制造（例如，在淬火），药物输送通过血管等方程描述这种变粘度流是不可压缩的Navier-Stokes方程与温度耦合。这些耦合问题的复杂性通常需要通过数值方法，通常是有限元方法来解决。然而，利息问题产生了两个主要困难。首先，这个问题是强耦合的，因为热量是由流体对流，而温度影响流动。此外，为了保证数值稳定性（和物理守恒性），流体速度必须是不可压缩的（以保证局部质量守恒）。此外，离散化导致一个非常大的非线性方程组，必须迭代求解，这使得高效的求解器成为必须。这两个困难是独立的，但又是相关的。更确切地说，在最近的工作之一的监督员（GRB）的一个新的战略来计算不可压缩的离散速度场，通过特设后处理的经典有限元方法，提出了。优点是众多的，但最重要的是恢复的物理特征的连续的解决方案，通常需要非常精细的网格近似。新的方法恢复这些功能的网格是相当粗糙，因此更容易计算。然而，不可压缩性约束必须完全满足，这意味着线性系统的某些方程必须非常精确地求解（直到机器精度）。这有时可以通过直接方法实现，但是当问题变得很大时，需要迭代方法，这通常不能高精度地求解方程。因此，迭代方法已被链接到一些数值实验描述的现实情况下，特别是在非常大的瑞利数的不准确性。

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.