Fast Integral Equation Methods: Algorithms and Applications

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The overall theme of the proposed research program is to develop fast and accurate integral equation methods for the purposes of investigating applications arising in fluid dynamics or other physical and biological systems. Numerical methods based on integral equations have become increasingly popular, due in large part to the development of associated fast algorithms that can be used to accelerate the solution procedure. In very general terms, a fast integral equation method (FIEM) first requires formulating a well-conditioned integral equation for a partial differential equation (PDE), then selecting a suitable quadrature method for its discretization, and finally implementing a fast algorithm, such as the fast multipole method or a fast direct solver, to accelerate the solution of the resulting linear system. The advantages of FIEMs over difference-based methods are significant: efficiency is obtained through dimension reduction and the use of fast algorithms; the ill-conditioning associated with directly discretizing the governing equation is avoided; and high-order accuracy is easier to attain. The computational efficiency of FIEMs can mean being able to solve a complex problem on a desktop computer in a matter of minutes instead of taking hours on a cluster. The superior stability properties coupled with high-order accuracy means that near machine-precision accuracy can be achieved. This high precision can be invaluable when using these tools for investigating analytic properties of PDEs, resolving complex features in solutions and providing benchmark data for other computational methods.**The two long-term goals discussed in the proposal are the following:**1. FIEM for Incompressible Fluid Dynamics*The incompressible Navier-Stokes equations (INSE) are a system of highly nonlinear, complex PDEs that are ubiquitous in describing a wide range of fluid phenomena, from swimming microorganisms to weather systems. While the INSE are not directly amenable to solution via integral equations, a suitable temporal discretization will yield a collection of linear, high-order elliptic equations that must be solved at each time step. The proposed research will develop methods to recast these equations as integral equations, by representing the solution as the sum of suitably chosen layer and volume potentials. Then, suitable fast algorithms will be used to solve for and evaluate these potentials. Significant progress has been made in developing the tools needed for a two-dimensional solver. We will also consider problems in three dimensions. **2. FIEM for Boundary Value Problems on Surfaces.*Applications involving the solution to PDEs on surfaces include computational fluid dynamics for planetary-scale flows, image processing, electromagnetic scattering, and pattern formation in biological systems. Current state-of-the-art methods do not include integral-equation based solvers. However, recent work on developing a fast-multipole accelerated solver for the Laplace-Beltrami equation for complex sub-manifolds on the surface of a sphere indicates that this is a very promising avenue of exploration.