Extending the applicability of integral equation methods: function extension and time dependent domains in 3D

The proposed research aims to extend the applicability of integral equation methods in 3D to enable accurate simulation of interacting viscous drops with soluble surfactants.

A boundary integral formulation is an excellent starting point to design numerical methods that accurately capture the interface dynamics for viscous drops in Stokes flow.

To include the effect of soluble surfactants on the dynamics, an advection-diffusion equation needs to be solved with boundary conditions at the drop surface.

This is not an elliptic PDE, and to take advantage of a boundary integral formulation to accurately enforce conditions at time-dependent boundaries, additional methodology is needed.The methodology proposed for solving the advection-diffusion equation on time dependent domains in 3D builds on elliptic marching, i.e. the solution of a sequence of inhomogeneous elliptic problems once the problem has been discretized in time.

This further involves a split into a particular and a homogeneous solution, the development of a method for extending a function with controllable global regularity, a framework for using this extension method within an adaptive time-stepping scheme on a time dependent domain, and special quadrature and fast summation methods to accurately solve a homogeneous elliptic equation to enforce the boundary conditions.

Once completed, this solver will be coupled to available software to allow also for soluble surfactants when simulating interacting viscous drops.