In traditional numerical methods for partial differential equations (e.g. finite-difference, finite-element methods), the equation is discretized and the resulting sparse linear system is solved by multigrid or Krylov iteration. Spectral methods, another well known approach, express the solution as a linear combination of orthogonal basis functions, and the fast Fourier transform (FFT) is used to compute the expansion coefficients. Despite algorithmic advances such as preconditioning and adaptive mesh refinement, these methods still struggle with problems involving complex geometry and multiscale features.

This project develops alternative methods in which the differential equation is first converted into an integral equation by convolution with the Green’s function, followed by discretization and linear solution. In the past these methods were hindered by the difficulty of discretizing singular integrals, and the cost of computing dense matrix-vector products, but advances in numerical analysis are bringing these obstacles under control. The research team believes that integral equation based methods are potentially superior, and the team aims to demonstrate this in regimes that are challenging for traditional methods.