A Boundary Integral Equation Method For The Laplace Equation With Dynamic Boundary Conditions

In this paper,we present a numerical algorithm for the accurate and efficient solution of the Laplace equation with dynamic boundary conditions on two dimensional complex geometry,which represents a primitive model for the moving contact lines,electro-wetting,and more general interactions of fluids with solid boundaries.We reformulate the partial differential equation to a time-dependent boundary integral equation using layer potentials where the unknowns are only defined on the boundary of the domain,and study the stability of the numerical discretization and its efficient solution.In particular,we show how existing state-of-the-art techniques for near-singular and singular integrals can significantly change the high frequency spectrum of the integral operators,which will cause numerical instability and unphysical solutions in the dynamic simulations.We present techniques to avoid the instability,and justify our analysis and demonstrate the accuracy and efficiency of the algorithm through several numerical experiments.