Fast integral equation methods for moving boundary problems in parabolic PDEs

快速算法，如快速多极子方法，小波，或基于FFT的卷积是目前公认的方法，用于求解椭圆型偏微分方程的边界积分公式。 这些方法已成功地应用于位势理论、粘性流动、线弹性和波传播等领域。本项目旨在将快速边界积分求解器的范围扩大到抛物型微分方程问题。在这种情况下，积分运算符涉及对问题历史的卷积，除非使用快速方法，否则会将CPU和存储要求提高到不切实际的水平。 具体来说，我们将研究Nystrom方法的边界积分方程的离散化，并开发基于热内核的Chebyshev插值的快速算法。我们希望在接近最佳的时间评估热层的潜力。 数值方法可以应用于热方程或瞬态Stokes流的问题。由于无条件的稳定性和更好的渐近缩放这些方法有可能取代传统的有限元或有限差分方法作为workhorsealgorithm.The新开发的数值技术将被应用到激光熔化的金属薄膜在基板上的问题，并与微和纳米结构表面的流体界面的相互作用。 这些问题对于技术应用是当前感兴趣的。激光诱导熔化金属薄膜的模拟将导致芯片实验室设备的微通道和纳米通道制造的潜在进展。这种便携式装置可用于检测从化学污染物到生物武器的一系列物质。流体界面与微米和纳米结构表面的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.