This dissertation presents fast integral equation methods (FIEMs) for solving two important problems encountered in practical engineering applications. The first problem involves the mixed boundary value problem in two-dimensional Stokes flow, which appears commonly in computational fluid mechanics. This problem is particularly relevant to the design of microfluidic devices, especially those involving superhydrophobic (SH) flows over surfaces made of composite solid materials with alternating solid portions, grooves, or air pockets, leading to enhanced slip. The second problem addresses waveguide devices in two dimensions, governed by the Helmholtz equation with Dirichlet conditions imposed on the boundary. This problem serves as a model for photonic devices, and the systematic investigation focuses on the scattering matrix formulation, in both analysis and numerical algorithms. This research represents an important step towards achieving efficient and accurate simulations of more complex photonic devices with straight waveguides as input and output channels, and Maxwell's equations in three dimensions as the governing equations. Numerically, both problems pose significant challenges due to the following reasons. First, the problems are typically defined in infinite domains, necessitating the use of artificial boundary conditions when employing volumetric methods such as finite difference or finite element methods. Second, the solutions often exhibit singular behavior, characterized by corner singularities in the geometry or abrupt changes in boundary conditions, even when the underlying geometry is smooth. Analyzing the exact nature of these singularities at corners or transition points is extremely difficult. Existing methods often resort to adaptive refinement, resulting in large linear systems, numerical instability, low accuracy, and extensive computational costs. Under the hood, fast integral equation methods serve as the common engine for solving both problems. First, by utilizing the constant-coefficient nature of the governing partial differential equations (PDEs) in both problems and the availability of free-space Green's functions, the solutions are represented via proper combination of layer potentials. By construction, the representation satisfies the governing PDEs within the volumetric domain and appropriate conditions at infinity. The combination of boundary conditions and jump relations of the layer potentials then leads to boundary integral equations (BIEs) with unknowns defined only on the boundary. This reduces dimensionality of the problem by one in the solve phase. Second, the kernels of the layer potentials often contain logarithmic, singular, and hypersingular terms. High-order kernel-split quadratures are employed to handle these weakly singular, singular, and hypersingular integrals for self-interactions, as well as nearly weakly singular, nearly singular, and nearly hypersingular integrals for near-interactions and close evaluations. Third, the recursively compressed inverse preconditioning (RCIP) method is applied to treat the unknown singularity in the density around corners and transition points. Finally, the celebrated fast multipole method (FMM) is applied to accelerate the scheme in both the solve and evaluation phases. In summary, high-order numerical schemes of linear complexity have been developed to solve both problems often with ten digits of accuracy, as illustrated by extensive numerical examples.
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