Spectral Accuracy in Fast Ewald Methods and Topics in Fluid Interface Simulation

This work contains two separate but related parts: one on spectrally accurate and fast Ewald methods for electrostatics and viscous flow, and one on micro- and complex fluid interface problems. In Part I we are concerned with fast and spectrally accurate methods to compute sums of slowly decaying potentials over periodic lattices. We consider two PDEs: Laplace (electrostatics, the Coulomb potential) and Stokes (viscous flow, the ``Stokeslet'' potential). Moreover, we consider both full and planar periodicity, the latter meaning that periodicity applies in two dimensions and the third is ``free''. These are major simulation tasks in current molecular dynamics simulations and in many areas of computational fluid mechanics involving e.g. particle suspensions. For each of the four combinations of PDE and periodic structure, we give spectrally accurate and O(N log N) fast methods based on Ewald's or Ewald-like decompositions of the underlying potential sums. In the plane-periodic cases we derive the decompositions in a manner that lets us develop fast methods. Associated error estimates are developed as needed throughout. All four methods can be placed in the P3M/PME (Particle Mesh Ewald) family. We argue that they have certain novel and attractive features: first, they are spectral accurate; secondly, they use the minimal amount of memory possible within the PME family; third, each has a clear and reliable view of numerical errors, such that parameters can be chosen wisely. Analytical and numerical results are given to support these propositions. We benchmark accuracy and performance versus an established (S)PME method. Part II deals with free boundary problems, specifically numerical methods for multiphase flow. We give an interface tracking method based on a domain-decomposition idea that lets us split the interface into overlapping patches. Each patch is discretized on a uniform grid, and accurate and efficient numerical methods are given for the equations that govern interface transport. We demonstrate that the method is accurate and how it's used in immersed boundary, and interface, Navier-Stokes methods, as well as in a boundary integral Stokes setting. Finally, we consider a problem in complex fluidics where there is a concentration of surfactants \emph{on} the interface and the interface itself is in contact with a solid boundary (the contact line problem). We argue that the domain-decomposition framework is attractive for formulating and treating complex models (e.g. involving PDEs on a dynamic interface) and proceed with developing various aspects of such a method.QC 2011112