An active area of research in our group is the development of fast and adaptive algorithms for computational problems in electromagnetics, materials science, device physics and chemistry. At the core of many simulations in these areas are solvers for the Poisson, Helmholtz, and time-harmonic Maxwell equations, time-domain wave propagation, and the diffusion equation.

While standard numerical methods typically rely on finite difference or finite element discretization of the governing partial differential equation, we concentrate on integral equation-based methods. These have a number of advantages - they are geometrically flexible, easy to use, compatible with embedded boundary discretizations , and lead to well-conditioned linear systems. They do, however, require some knowledge of the underlying Green's function and a variety of fast algorithms in order to be practical.

Fast Multipole Methods

The simplest example of an "integral-equation" based fast solver is, perhaps, the recasting of the Poisson equation with a right-hand side consisting of a set N point sources (delta functions) located at {P

- L. Greengard and V. Rokhlin,
*A Fast Algorithm for Particle Simulations*, J. Comput. Phys.**73**, 325-348 (1987).

- H. Cheng, L. Greengard and V. Rokhlin,
*A Fast Adaptive Multipole Algorithm in Three Dimensions*, J. Comput. Phys.**155**, 468-498 (1999).

For recent technical references on extensions to Stokes, Helmholtz and screened Coulomb interactions, see:

- Z. Gimbutas and L. Greengard,
*Fast Multi-Particle Scattering: a Hybrid Solver for the Maxwell Equations in Microstructured Materials,*J. Comput. Phys.,**232**, 22-32 (2013). - A.-K. Tornberg and L. Greengard,
*A Fast Multipole Method for the Three Dimensional Stokes Equations*, J. Comput. Phys.**227**, 1613-1619 (2008). - H. Cheng, W. Y. Crutchfield, Z. Gimbutas, L. Greengard,
F. Ethridge, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao,
*A Wideband Fast Multipole Method for the Helmholtz Equation in Three Dimensions*, J. Comput. Phys.**216**, 300-325 (2006). - L. Greengard and J. Huang,
*A New Version of the Fast Multipole Method for Screened Coulomb Interactions in Three Dimensions*, J. Comput. Phys.**180**, 642-658 (2002).

FMM and related algorithms are not limited to particle interactions. They are used in accelerating integral-equation based solvers for many of the partial differential equations of classical physics.