Fast Solvers for Stokes Flow past Axisymmetric Geometries
At very small length scales (e.g. for describing motions of bacterias), the motion of a fluid is approximated by the Stokes equations. These elliptic partial differential equations admit Greens functions, which enable writing the solution of a boundary value problem as a boundary integral equation (BIE). It is well established that using BIE formulation, we can construct a more accurate solution faster compared to traditional finite difference/volume/element schemes.
Nevertheless, due to the relatively poor quality of two-dimensional singular quadratures, we require a high level of discretization to achieve high accuracy for general surface boundaries. In this project, we assume that these surfaces are axisymmetric, and leverage this rotational symmetry to decouple 2D BIE into a series of 1D BIEs. Using the much better quality 1D quadrature schemes, we can easily achieve high accuracy. In addition, the decoupling improves the computational complexity of our overall solver.
At the end of the project, we aim to implement our solution algorithm, and couple it with fast multipole method (FMM) to construct a framework for computing Stokes flow past a large number of axisymmetric bodies.