Application of the eXtended finite element method in mathematical biology.

摘要

Across a broad spectrum of mathematical models, problems arise that have irregular domains or contain embedded interfaces. The complexity due to the introduction of these interfaces makes it more difficult to develop efficient and accurate numerical methods for their solutions.; This thesis discusses modifications to the eXtended Finite Element Method (X-FEM) and its application to problems involving embedded interfaces. The X-FEM is shown to be globally second order accurate and is convergent for both the solution and the gradient of the solution on the interface. In addition, the {lcub}X-FEM{rcub} is formulated for both steady and time dependent nonlinear partial differential equations.; Applications of the X-FEM are given for steady Stokes and Navier-Stokes flows over obstacles as well as a study of quorum sensing in bacterial biofilm for various biofilm configurations. A bacterial biofilm is a collection of bacteria attached to a surface by the means of an extracellular polymer. Through a process called quorum sensing, bacteria contained in a biofilm monitor their population density by using extracellular signaling molecules. Quorum sensing is modeled using a 2-D time-dependent nonlinear reaction-advection-diffusion equation in a domain containing an irregular embedded interface. In addition, the nonlinear source term is discontinuous in the unknown variable. The X-FEM handles this model accurately as well as computing the fluid flow over the biofilm surface that drives the advection of the signaling molecules.