Robust preconditioning of multiphysics problems and interstitial fluid flow

Abstract

In the natural sciences, there are many examples of physical systems which can be modeled by differential equations, such as the flow of water through soil or the deformation of a bridge under load. By developing mathematical techniques for solving the resulting equations with a computer, we can use these models to answer questions such as how much weight a bridge can support before collapsing. Complex systems often involve multiple interdependent kinds of physical behavior. An example is the brain, where water movement occurs both inside blood vessels and in the surrounding brain tissue. Because the two kinds of flow affect each other through interchange across the vessel walls, a model of water movement in the brain must account for both. However, even when there are methods for solving either subproblem individually, coupling them together in a robust manner is not always straightforward, and the focus of this thesis. Robustness means that the method should run in a reasonable amount of time, both for complex problems with many unknowns and for all values of the problem parameters, even the very large or very small. This property is crucial for applications in biomechanics.