Implicit Boundary Integral Method for Homogeneous Hele-Shaw Problem with multi-connected Domain

Abstract

In this work, we implement the implicit boundary integral method for a homogeneous Hele-Shaw problem with a multi-connected domain. This method base on the solution of layer potential integral for the Laplace equation. The numerical technique is easy to implement, base on the idea of averaging the parameterization near the boundary and applying the Coarea formula. This technique changes the boundary integral into the Riemann integral that numerically easy to compute. The difficulty in the computation of hypersingular integral occurs to compute the normal velocity of free boundary. We use a collocation technique to eliminate the hypersingular part in the integral equation. Also, we show the numerical results and its computation performance due to the appearance of a non-invertible matrix.