A multigrid method for kernel functions acting on interacting structures with applications to biofluids

Abstract

Simulating the dynamics of discretized interacting structures whose relationship is dictated by a kernel function gives rise to a large dense matrix. We propose a multigrid solver for such a matrix that exploits not only its data-sparsity resulting from the decay of the kernel function but also the regularity of the geometry of the structures and the quantities of interest distributed on them. Like the well-known multigrid method for large sparse matrices arising from boundary-value problems, our method requires a smoother for removing high-frequency terms in solution errors, a strategy for coarsening a grid, and a pair of transfer operators for exchanging information between two grids. We develop new techniques for these processes that are tailored to a kernel function acting on discretized interacting structures. They are matrix-free in the sense that there is no need to construct the large dense matrix. Numerical experiments on a variety of bio-inspired microswimmers immersed in a Stokes flow demonstrate the effectiveness and efficiency of the proposed multigrid solver. In the case of free swimmers that must maintain force and torque balance, additional sparse rows and columns need to be appended to the dense matrix above. We develop a matrix-free fast solver for this bordered matrix as well, in which the multigrid method is a key component.