Michael Siegel

Michael Siegel, Anna-Karin Tornberg

Accurate evaluation of layer potentials is crucial when boundary integral equation methods are used to solve partial differential equations. Quadrature by expansion (QBX) is a recently introduced method that can offer high accuracy for singular and nearly singular integrals, using truncated expansions to locally represent the potential. The QBX method is typically based on a spherical harmonics expansion which when truncated at order $p$ has $O(p^2)$ terms. This expansion can equivalently be written with $p$ terms, however paying the price that the expansion coefficients will depend on the evaluation/target point. Based on this observation, we develop a target specific QBX method, and apply it to Laplace's equation on multiply connected domains. The method is local in that the QBX expansions only involve information from a neighborhood of the target point. An analysis of the truncation error in the QBX expansions is presented, practical parameter choices are discussed and the method is validated and tested on various problems.

Sara Pålsson, Michael Siegel, Anna-Karin Tornberg

Performing highly accurate simulations of droplet systems is a challenging problem. This is primarily due to the interface dynamics which is complicated further by the addition of surfactants. This paper presents a boundary integral method for computing the evolution of surfactant-covered droplets in 2D Stokes flow. The method has spectral accuracy in space and the adaptive time-stepping scheme allows for control of the temporal errors. Previously available semi-analytical solutions (based on conformal-mapping techniques) are extended to include surfactants, and a set of algorithms is introduced to detail their evaluation. These semi-analytical solutions are used to validate and assess the accuracy of the boundary integral method, and it is demonstrated that the presented method maintains its high accuracy even when droplets are in close proximity.