An accurate integral equation method for Stokes flow with piecewise smooth boundaries

Abstract

Two-dimensional Stokes flow through a periodic channel is considered. The channel walls need only be Lipschitz continuous, in other words they are allowed to have corners. Boundary integral methods are an attractive tool for numerically solving the Stokes equations, as the partial differential equation can be reformulated into an integral equation that must be solved only over the boundary of the domain. When the boundary is at least C 1 smooth, the boundary integral kernel is a compact operator, and traditional Nyström methods can be used to obtain highly accurate solutions. In the case of Lipschitz continuous boundaries, however, obtaining accurate solutions using the standard Nyström method can require high resolution. We adapt a technique known as recursively compressed inverse preconditioning to accurately solve the Stokes equations without requiring any more resolution than is needed to resolve the boundary. Combined with a periodic fast summation method we construct a method that is O(N log N) where N is the number of quadrature points on the boundary. We demonstrate the robustness of this method by extending an existing boundary integral method for viscous drops to handle the movement of drops near corners.

A Unified Formulation of Analytical and Numerical Methods for Solving Linear Fredholm Integral Equations

“…In this Section, we explore the use of some of the numerical integration techniques to approximate the integral operator in (1) and to thus obtain a semi-discrete equation of the kind (2).…”

“…Fredholm integral equations arise in the mathematical modeling of various processes in science and engineering, but also as reformulations of differential boundary value problems in applied mathematics. For example, in [1], a two-dimensional Stokes flow through a periodic channel problem is reformulated into an integral equation over the boundary of the domain and solved numerically; in [2], the solution of several boundary value problems and initial boundary value problems of interest to geomechanics through their reduction to integral equations is described, and many related references are cited; in [3], several different approaches to transformation of the second-order ordinary differential equations into integral equations is presented, and approximate solutions are derived via numerical quadrature methods; in [4], planar problems for Laplace's equation are reformulated as boundary integral equations and then solved numerically.…”

“…In this paper, for simplicity, we confine our investigations to one dimension with x ∈ [a, b] ⊂ R, but the results obtained can be extended to other regions in two or more dimensions. Integral equations of the type (1) have been studied by many researchers over the last century and continue to receive much attention in recent years. For their solution, a variety of methods have been developed; see [5][6][7][8][9] and others.…”

“…Here, we present a common formulation suitable for symbolic computations for all of these techniques. Our approach is based on the idea that in all instances, the integral equation in (1) may be cast or reduced to an equation of the form…”

“…Both {g j (x)} and {Ψ j } are obtained through the separation or approximation of the kernel, the approximation of the integral, the unknown function or a combination of them. Equation ( 2), under certain conditions, which are associated with the existence and uniqueness of the solution of the integral equation in (1), can be solved symbolically to obtain an exact or approximate analytic solution of (1). We implement the proposed method to construct exact closed-form solutions when the kernel K(x, t) is separable, approximate analytic solutions when K(x, t) is not separable, but it can be represented as a truncated power series or an interpolation polynomial, and semi-discrete solutions when the definite integral is replaced by a finite sum by using a quadrature rule.…”