Boundary integral equations (BIE) are widely used for simulating systems in viscous or small scale fluids modeled by the Stokes equation. Often, systems of interest live in confined and/or periodic geometries, such as the transport of cells in the blood vessel or bacterial suspensions near a stationary wall. The study of these systems benefit from the dimension reduction of the BIE methods, but computation costs need to be reduced further due to the multiscale behaviors and large degrees of freedom. The free-space fundamental solution is often used in the BIE formulation, but having a fundamental solution that satisfies more boundary conditions means less computation during the BIE solve.

In this talk I will present the fundamental solution for the 3D half-space, where a stationary wall is set at z=0 with no-slip boundary conditions [1]. I will then present an alternative formulation by [2] that is individually net-force-free for periodization. These formulations use the free-space fundamental solutions of Stokes and Laplace equations and their derivatives, and are therefore easily implemented and optimized due to the many free-space fast algorithms available, including FMM.

[1] Z. Gimbutas, L. Greengard, S. Veerapaneni, Simple and efficient representations for the fundamental solutions of Stokes flow in a half-space, J. Comput. Mech., 776, 2015
[2] W. Yan, M. Shelley, Universal image systems for non-periodic and periodic Stokes flows above a no-slip wall, J. Comput. Phy., 375:263-270, 2018