A center of mass method with applications to the solution of the two-dimensional Stokes equations in a channel.

摘要

In this work we consider the Stokes equations{dollar}{dollar}cases{lcub}qquadqquadenspacenablacdot u=0crcr {lcub}-{rcub}nabla p+munablasp2u+gdelta(x-xsb0)=0cr{rcub}{dollar}{dollar}in a two dimensional bounded domain {dollar}Omega.{dollar} These equations arise from the study of two dimensional drops suspended in a fluid.; There are a variety of methods to solve these equations. Two proven methods are the finite element method and the boundary integral method. We consider the boundary integral method to approximate the velocity field for the suspended drops. The kernels of the derived integral equations are the Green's function and its associated stress tensor for the Stokes flow.; Approximating the Green's function is necessary in order to obtain an efficient method in computing the integral equations. An adaptive center of mass method, based on the data structures of the fast multipole method formulated by L. Greengard, is derived.; For the bounded domain, the Green's function is separated into a singular function and an analytic function. The center of mass method gives a "fast" algorithm to approximate the singular part of the Green's function. An adaptive strategy, based on Hermite interpolants is used to approximate the analytic function (which is dependent on the four independent variables of the two-dimensional points).; Error estimates are given for the approximation using the center of mass method. Based on the estimates, an algorithm is derived to obtain a required a priori bound on the approximation error. Defining the near and the far field is dependent on the error requirements for each problem.; The strategy of approximation by the centers of mass of the far field particles creates a robust method for approximating singular functions which are not harmonic.