An algorithm for the direct inversion of the linear systems arising from Nyström discretization of integral equations on one-dimensional domains is described. The method typically has O(N ) complexity when applied to boundary integral equations (BIEs) in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes' equations. The scaling coefficient suppressed by the "big-O" notation depends logarithmically on the requested accuracy. The method can also be applied to BIEs with oscillatory kernels such as those associated with the Helmholtz and Maxwell equations; it is efficient at long and intermediate wave-lengths, but will eventually become prohibitively slow as the wave-length decreases. To achieve linear complexity, rank deficiencies in the off-diagonal blocks of the coefficient matrix are exploited. The technique is conceptually related to the H-and H 2matrix arithmetic of Hackbusch and co-workers, and is closely related to previous work on Hierarchically Semi-Separable matrices.

Abstract. Boundary integral equations and Nyström discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weaklysingular kernel arises, in which case specialized quadratures that modify the matrix entries near the diagonal are needed to reach a high accuracy. We describe the construction of four different quadratures which handle logarithmically-singular kernels. Only smooth boundaries are considered, but some of the techniques extend straightforwardly to the case of corners. Three are modifications of the global periodic trapezoid rule, due to Kapur-Rokhlin, to Alpert, and to Kress. The fourth is a modification to a quadrature based on Gauss-Legendre panels due to Kolm-Rokhlin; this formulation allows adaptivity. We compare in numerical experiments the convergence of the four schemes in various settings, including low-and high-frequency planar Helmholtz problems, and 3D axisymmetric Laplace problems. We also find striking differences in performance in an iterative setting. We summarize the relative advantages of the schemes.

A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in R 3 is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a surface to a sequence of BIEs defined on a generating curve for the surface. It can handle loads that are not necessarily rotationally symmetric. Nyström discretization is used to discretize the BIEs on the generating curve. The quadrature is a high-order Gaussian rule that is modified near the diagonal to retain highorder accuracy for singular kernels. The reduction in dimensionality, along with the use of high-order accurate quadratures, leads to small linear systems that can be inverted directly via, e.g., Gaussian elimination. This makes the scheme particularly fast in environments involving multiple right hand sides. It is demonstrated that for BIEs associated with the Laplace and Helmholtz equations, the kernel in the reduced equations can be evaluated very rapidly by exploiting recursion relations for Legendre functions. Numerical examples illustrate the performance of the scheme; in particular, it is demonstrated that for a BIE associated with Laplace's equation on a surface discretized using 320 800 points, the set-up phase of the algorithm takes 1 minute on a standard laptop, and then solves can be executed in 0.5 seconds.

This paper examines the electrostatic force on a microdroplet transported via electrowetting on dielectric (EWOD). In contrast with previous publications, this article details the force distribution on the advancing and receding fluid faces, in addition to presenting simple algebraic formulae for the net force in terms of system parameters. Dependence of the force distribution and its integral on system geometry, droplet location, and material properties is described. The consequences of these theoretically and numerically obtained results for design and fabrication of EWOD devices are considered.

Two primary methods for electrostatically actuating microdroplets in channels currently exist: dielectrophoresis (DEP) for electrically insulating fluids and electrowetting on dielectric (EWOD) for conducting fluids. In each case, a transverse electric field is used to create an electrostatic pressure, giving rise to the transport of individual liquid slugs. This paper examines the nature of the force distribution for both EWOD and DEP actuated droplets. The effects of system parameters such as contact angle and electrode length on the shape of the force density and its net integral are considered. A comparison of the scaling properties of DEP and EWOD for applications in digital microfluidics is presented. The net DEP force is shown to be strongly peaked when a droplet interface is located near the edge of a charged electrode and reduces to the well-known lumped parameter model in the appropriate limits. The effect of electrode spacing is seen to have an inversely proportional effect on the force experienced by the droplet, and the effect of increasing droplet contact angle is observed to increase the net force on the droplet.

a b s t r a c tA numerical method for solving the equations modeling acoustic scattering in three dimensions is presented. The method is capable of handling several dozen scatterers, each of which is several wave-lengths long, on a personal work station. Even for geometries involving cavities, solutions accurate to seven digits or better were obtained. The method relies on a Boundary Integral Equation formulation of the scattering problem, discretized using a high-order accurate Nyström method. A hybrid iterative/direct solver is used in which a local scattering matrix for each body is computed, and then GMRES, accelerated by the Fast Multipole Method, is used to handle reflections between the scatterers. The main limitation of the method described is that it currently applies only to scattering bodies that are rotationally symmetric.

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Transporting microdroplets using electric fields can be accomplished with several mechanisms, the primary methods being dielectrophoresis (DEP) for electrically insulating liquids, and electrowetting on dielectric for conducting fluids. In both cases, an electric field is applied near the leading edge of the droplet using patterned electrodes, giving rise to an electrostatic pressure that induces droplet transport. This paper examines the nature of the force distribution for DEP-actuated droplets in several electrode configurations, calculated using a numerical method designed for handling jump conditions in the Poisson equation. The numerical method is described and verified by comparison with known analytical results. The net force acting upon a DEP droplet is investigated, with the effect of electrode configuration presented for several cases, demonstrating some beneficial aspects for engineering applications.