Fast integral equation methods for moving boundary problems in parabolic PDEs

快速算法，如快速多极方法、小波或基于fft的卷积，是目前解决椭圆型偏微分方程边界积分公式的有效方法。这些方法已成功地应用于势理论、粘性流动、线弹性和波传播。该项目旨在扩大快速边界积分求解器的范围，以解决由抛物型微分方程控制的问题。在这种情况下，积分运算符涉及到对问题历史的卷积，这将使CPU和存储需求增加到不切实际的水平，除非使用快速方法。具体来说，我们将研究边界积分方程离散化的Nystrom方法，并开发基于热核切比雪夫插值的快速算法。我们期望在接近最佳的时间内评估热层势。数值方法可以应用于热方程或瞬态斯托克斯流控制的问题。由于这些方法具有无条件稳定性和更好的渐近标度，因此有可能取代传统的有限元或有限差分方法作为主力算法。新发展的数值技术将应用于激光熔化基板上金属薄膜的问题，以及流体界面与微纳米结构表面的相互作用。这些问题是目前技术应用所关心的。激光诱导金属薄膜熔化的模拟将导致微纳米通道在芯片实验室器件制造中的潜在进展。这种便携式设备可用于检测从化学污染物到生物武器的一系列物质。流体界面与微观和纳米结构表面相互作用的模拟对于所谓的自清洁表面的发展是重要的，这被认为是纳米技术最有前途的应用之一。拟议的项目还将通过研究和跨学科科学合作促进学习。

Fast algorithms, such as the Fast Multipole Method, wavelets, or FFT-based convolutions are by now well established methods for solving boundary integral formulations of elliptic partial differential equations. These methods have been successfully applied in potential theory, viscous flow, linear elasticity and wave propagation.This project seeks to broaden the range of fast boundary integral solvers to problems governed by parabolic differential equations. In this case the integral operators involve a convolution over the history of problem, which increase CPU and storage requirements to impractical levels unless fast methods are used. Specifically, we will investigate Nystrom methods for the discretization of boundary integral equations and develop fast algorithms based on Chebyshev interpolation of the heat kernel. We expect to evaluate thermal layer potentials in nearly optimal time. The numerical methods can be applied to problems governed by the heat equation or transient Stokes flow. Because of unconditional stability and better asymptotic scaling these methods have the potential to replace the conventional finite element- or finite difference methods as the workhorse algorithm.The newly developed numerical techniques will be applied to the problem of laser melting of a metal film on a substrate and to the interaction of fluid interfaces with micro- and nanostructured surfaces. These problems are of current interest for technological applications. The simulations of laser-induced melting of metal films will lead to potential advances in fabrication of micro- and nanochannels for lab-on-a-chip devices. Such portable devices can be used for detection of a range of substances from chemical pollutants to biological weapons. The simulations of interaction of fluid interfaces with micro- and nanostructured surfaces are important for the development of the so-called self-cleaning surfaces, which are considered among the most promising applications of nanotechnology. The proposed project will also promote learning through research and interdisciplinary scientific collaborations.