CAREER: Towards General-Purpose, High-Order Integral Equation Methods for Computer Simulation in Engineering: Analysis, Algorithm Design, and Applications

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Numerical simulation has become an essential tool in nearly all areas of science and engineering, ranging from engine design to naval architecture, and from personalized medicine to city planning. Yet, the efficient solution of large-scale, globally coupled (so-called elliptic) computational problems arising in these application areas remains a major challenge. Although integral equation (IE) methods for numerical simulation typically have optimally low cost when applied to common problems in science and engineering, their impact has been limited to a small handful of applications by technical obstacles. The purpose of this project is to take important steps to remove these obstacles. First, the project will develop novel numerical and symbolic algorithms to reduce the amount of method design and implementation work required when adapting IE methods to new classes of applications. This will make the associated cost savings of these methods more broadly accessible. Second, the project will design, implement, and analyze parallel algorithms to enable the nation's large-scale computing resources to be used in conjunction with IE methods. Their increased computational efficiency will facilitate the study of models with increased fidelity and higher accuracy. Third, the project will extend the set of problems that can be attacked with IE methods to those including volume (and not just surface) data while using highly accurate geometric representations. Fourth, it will provide a theoretical understanding and practical methods for automatic control of numerical error in these methods. Lastly, the project will demonstrate the new methods and their use in the mathematically and numerically challenging context of fluid dynamics. To foster an understanding of the power of these kinds of computational tools in the next generation of the nation's workforce, this project will employ a day-long experience for students in their formative middle-school years. This experience will convey that computer modeling and simulation can help understand the world by testing the predictive power of simple, mechanistic models. Through its reliance on self-contained, hands-on computer experiments, the experience will be interactive and visually engaging, require little mathematics preparation, and easily establish connections with real-world applications of computing. The program focuses on creating engagement and interest, with the goal of promoting career and educational choices in mathematics and computing. Integral equation (IE) methods for computer simulation typically have optimally low cost, but their impact has been limited to a handful of applications by technical obstacles. The purpose of this research is to take important steps to remove these obstacles, by providing: 1. high-order singular quadrature and infrastructure for fast multipole methods for the evaluation of layer potentials with general, symbolically given kernels in complex geometry, 2. scalable and efficient distributed-memory parallel algorithms for the use of IE methods in large-scale applications, 3. design and analysis of high-order numerical methods for volume potentials in complex geometry as needed by inhomogeneous partial differential equations, 4. theory and methods for automatic adaptive mesh refinement based on a-posteriori error estimates, and 5. a demonstration of the capabilities of the developed methods and algorithms in the context of the incompressible Navier-Stokes equations, including high-order finite-element-method and IE coupling.