Ludvig af Klinteberg: Kernel-split fast summation of Stokes potentials

Abstract:

Kernel splitting refers to the decomposition of a fundamental solution, such as the \(1/r\) kernel of Laplace's equation, in a way that separates the short-range singularity from the smooth long-range interactions. Combined with Fourier methods this can be used to develop fast summation methods, such as fast Ewald summation which has \(O(N \log N)\) complexity. The recently introduced DMK (dual-space multilevel kernel-splitting) framework combines elements of this with the hierarchical structure of the Fast Multipole Method (FMM), leading to a very fast method with \(O(N)\) complexity.

In this talk we will discuss kernel-splitting of the common kernels of Stokes flow: the Stokeslet, the stresslet and the rotlet. Specifically, we will see how kernel-splits can be derived in Fourier space, and how this allows the use of highly efficient window functions known as prolate spheroidal wave functions (PSWFs). This then allows us to formulate the DMK methods for the Stokes kernels.