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This paper presents a new boundary integral equation (BIE) method for simulating particulate and multiphase flows through periodic channels of arbitrary smooth shape in two dimensions. The authors consider a particular system--multiple vesicles suspended in a periodic channel of arbitrary shape--to describe the numerical method and test its performance. Rather than relying on the periodic Green's function as classical BIE methods do, the method combines the free-space Green's function with a small auxiliary basis and imposes periodicity as an extra linear condition. As a result, we can exploit existing free-space solver libraries, quadratures, and fast algorithms and handle a large number of vesicles in a geometrically complex channel. Spectral accuracy in space is achieved using the periodic trapezoid rule and product quadratures, while a first-order semi-implicit scheme evolves particles by treating the vesicle-channel interactions explicitly. New constraint-correction formulas are introduced that preserve reduced areas of vesicles, independent of the number of time steps taken. By using two types of fast algorithms--(i) the fast multipole method for the computation of the vesicle-vesicle and the vesicle-channel hydrodynamic interaction, and (ii) a fast direct solver for the BIE on the fixed channel geometry--the computational cost is reduced to O(N) per time step, where N is the spatial discretization size. Moreover, the direct solver inverts the wall BIE operator at t = 0, stores its compressed representation, and applies it at every time step to evolve the vesicle positions, leading to dramatic cost savings compared to classical approaches. Numerical experiments illustrate that a simulation with N = 128;000 can be evolved in less than a minute per time step on a laptop. [ABSTRACT FROM AUTHOR]

Dense particulate flow simulations using integral equation methods demand accurate evaluation of Stokes layer potentials on arbitrarily close interfaces. In this paper, we generalize techniques for close evaluation of Laplace double-layer potentials in [J. Helsing and R. Ojala, J. Comput. Phys., 227 (2008), pp. 2899-2921]. We create a "globally compensated" trapezoid rule quadrature for the Laplace single-layer potential on the interior and exterior of smooth curves. This exploits a complex representation, a product quadrature (in the style of Kress) for the sawtooth function, careful attention to branch cuts, and second-kind barycentric-type formulae for Cauchy integrals and their derivatives. Upon this we build accurate single- and double-layer Stokes potential evaluators by expressing them in terms of Laplace potentials. We test their convergence for vesicle-vesicle interactions, for an extensive set of Laplace and Stokes problems, and when applying the system matrix in a boundary value problem solver in the exterior of multiple close-to-touching ellipses. We achieve typically 12 digits of accuracy using small numbers of discretization nodes per curve. We provide documented codes for other researchers to use. [ABSTRACT FROM AUTHOR]