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 L2(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.