Convergence of a Multiple Reflection Method for Calculating Stokes Flow in a Suspension
Jonathan H. C. Luke · SIAM Journal on Applied Mathematics · 1989
The calculation of solutions of Stokes equations in the complex geometry of a suspension with many particles of arbitrary shape and arrangement is reduced to a sequence of calculations of flows with single particle boundary conditions. The sequence of approximate solutions converge to the solution of the full problem in the energy dissipation norm at an exponential rate. The convergence proof is based on the observation that the flow in a suspension minimizes the rate of energy dissipation over a certain class of flows. Each flow in the approximating sequence minimizes the rate of dissipation over a subclass of these flows containing the previous flow, so the energy dissipation norm is decreasing monotonically. From the latter minimum principle, it follows that each approximate flow is obtained from the previous one through application of one of a finite number of orthogonal projection operators. The properties of the subspaces of flows associated with these projections assure that the sequence converges.