Exact Singularity Subtraction from Boundary Integral Equations in Modeling Vesicles and Red Blood Cells

The study of vesicles, capsules and red blood cells (RBCs) under flow is a field of active research, belonging to the general problematic of fluid/structure interactions. Here, we are interested in modeling vesicles, capsules and RBCs using a boundary integral formulation, and focus on exact singularity subtractions of the kernel of the integral equations in 3D. In order to increase the precision of singular and near-singular integration, we propose here a refinement procedure in the vicinity of the pole of the Green-Oseen kernel. The refinement is performed homogeneously everywhere on the source surface in order to reuse the additional quadrature nodes when calculating boundary integrals in multiple target points. We also introduce a multi-level look-up algorithm in order to select the additional quadrature nodes in vicinity of the pole of the Green-Oseen kernel. The expected convergence rate of the proposed algorithm is of order O(1/N-2) while the computational complexity is of order O(N-2 ln N), where N is the number of degrees of freedom used for surface discretization. Several numerical tests are presented to demonstrate the convergence and the efficiency of the method.