Abstract
This research project develops a mathematical and numerical framework for representing static fluid interfaces as embedded manifolds. A variational principle is developed for the embedding function of a smooth manifold, along with the necessary boundary and gauge conditions. The variational problem is solved by a combination of Finite Elements, constrained optimisation techniques, and original algorithms. The approach is applied to problems inspired by modern technological and scientific applications of static fluid interfaces, and the results are compared to exact solutions, experimental data, or other numerical methods, where possible. The impact of the numerical methods on the quality of the solution is discussed in de- tail, with reference to the boundary conditions, the gauge conditions, and the constraints.