Accurate near-contact simulation in 2D Stokes flow via two-body preconditioning

Dense suspensions of rigid particles in viscous flow are notoriously difficult to simulate. As particles approach one another, near-contact interactions generate extremely fine spatial scales and severe ill-conditioning, making standard numerical methods either inaccurate or prohibitively expensive to scale. In this talk, I will present a new two-body preconditioning strategy for efficiently resolving near-contact interactions. The method combines a coarse global representation with local high-resolution corrections computed only for nearby particle pairs. These pairwise corrections are precomputed, compressed into equivalent coarse representations, and incorporated into a global iterative solve involving only coarse degrees of freedom. The framework is general and compatible with a broad class of boundary value problem solvers, with potential applications beyond Stokes flow to other elliptic PDEs. I will demonstrate the approach through the method of fundamental solutions applied to two-dimensional Stokes mobility and resistance problems, including simulations of suspensions containing up to 10,000 disks. The resulting solver remains accurate even in close-to-touching configurations while achieving rapid convergence and near-linear scaling with problem size.