Potential theory for initial-boundary value problems of unsteady Stokes flow in two dimensions

Integral equations have been of great theoretical importance for analyzing boundary value problems. There is a large amount of literature devoted to the classical potential theory and its applications on solving the boundary value problems of elliptic partial differential equations (see, for example, [5, 20, 23, 25, 26, 31, 32, 36, 37, 40]). For elliptic problems, integral equations have been coupled with finite element methods in numerical computation and the resulting boundary element methods have been very popular in engineering science (see, for example, [2, 3, 24]). For time-dependent problems, integral equation methods are less successful in numerical computation. The primary reason is that the direct implementation of integral equation methods for time-dependent problems is computationally expensive as compared with finite difference or finite element methods. Indeed, the discretizations of integral equations usually lead to dense linear systems. And for time dependent problems, the layer potentials involve integration in both space and time, which makes the evaluation of layer potentials and time marching extremely expensive. The invention of the Fast Multipole Method (FMM) (see, for example, [16, 17, 4]) has dramatically changed the landscape of the field of scientific computing. Tremendous progress has been made in designing fast and accurate numerical algorithms using FMM and its descendents to solve integral equations for various problems in electromagnetics, elasticity, and fluid mechanics (see, for example, [6, 14, 12, 13, 33, 45, 46, 47]). The numerical tools for solving the heat equation using integral equations have also been developed recently (see, for example, [18, 19, 28, 15, 43, 44]). Hyperbolic potentials have also been applied to study time-dependent problems for scattering problems in electromagnetics (see, for example, [30]). When the hurdle of computational cost has been overcome, integral equation methods offer several advantages as compared with standard finite difference and/or finite element methods. First, problems of complex geometry can be dealt with more easily. Second, the artificial boundary conditions are avoided for exterior problems. Third, the dimension of the problem is reduced by one for certain problems. Fourth, the influences of the initial data, nonhomo-