| Problems involving irregularly shaped domains or embedded interfaces occur in a wide range of mathematical models. These include models in computational biology, fluid mechanics, and solidification. Numerical solutions for this type of problem can be difficult to obtain, and therefore a variety of methods have been devised to solve them. One method that has been shown to perform well in comparison with other methods is the eXtended Finite Element Method (X-FEM).;In this thesis, a variety of modifications are made to the X-FEM in order to apply it to problems with special interfacial features. First, a custom enrichment function is shown to increase the accuracy of the X-FEM for a problem containing a boundary layer at an internal interface. This technique is then applied to the problem of biofilm growth in order to explore the relationship between the shape of a biofilm colony and the profile of its growth. Next, the X-FEM is applied to film thinning in foam dynamics, and we present a method for solving the Stokes equations while accounting for surface tension along a free surface. Finally, we extend the model for film thinning to include a solidification front. The process of foam solidification is challenging to model computationally due to the presence of a triple junction where the solidification front intersects the gas-liquid interface. We propose two procedures for determining the subgrid location of the triple junction and demonstrate how to properly prescribe boundary conditions along each interface. This method is then applied to a simplified model for foam solidification in order to generate some preliminary results. |
