**Prof. Anna-Karin Tornberg **(KTH Royal Institute of Technology, Stockholm, Sweden)

**Highly accurate integral equation
based methods for surfactant laden drops in two and three dimensions.**

In micro-fluidics, at small scales where inertial
effects become negligible, surface to volume ratios are large and the
interfacial processes are extremely important for the overall dynamics.
Integral equation based methods are attractive for the simulations of e.g.
droplet-based microfluidics, with tiny water drops dispersed in oil, stabilized
by surfactants.

We have developed highly accurate numerical methods
for drops with insoluble surfactants, both in two and three dimensions. In this
talk I will discuss some fundamental challenges that we have addressed, that are
also highly relevant to other applications: accurate quadrature methods for
singular and nearly singular integrals, adaptive time-stepping, and
reparameterization of time-dependent surfaces for high quality discretization
of the drops throughout the simulations. I will also discuss a recent extension
to include electric fields, as well as quadrature error estimates in 2D and the
extension of such estimates to 3D.

**Dr Euan Spence (University of Bath)**

**Does the Galerkin method converge
for the standard second-kind integral equations for the Laplacian on Lipschitz
domains?**

It has not yet been proved that the Galerkin method
converges when applied to the standard second-kind integral-equation
formulations for Laplace’s equation on general Lipschitz domains, or even
general 3D Lipschitz polyhedra.

This convergence result is equivalent to proving that
the relevant integral operators are the sum of a coercive operator and a
compact operator on L^{2}(Γ), where Γ denotes
the boundary of the Lipschitz domain.

In this talk, I will describe recent results obtained
with Simon Chandler-Wilde (University of Reading) that settle this question**. **

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