Integral Equation Methods for Variable Coefficient Elliptic Problems and Applications

本建议的重点是数学分析和有效实现基于积分方程的方法（IEM），以解决物理上重要的，空间上非齐次或“变系数”椭圆方程。这些出现在需要模拟复杂流体流动的应用中，例如燃烧或地下水污染。它们也出现在新型流体微机电系统（MEMS）以及固态半导体器件的设计中。在过去的十年中，iem已经被证明是非常成功的，并且现在被广泛应用于电磁、弹性和流体动力学建模中，当控制方程是“常系数”型时。为了解决更一般的变系数问题，我们使用格林函数将解表示为一个体积积分，用于一个简单的，与未知密度卷积的邻近问题。将这个表达式代入原来的椭圆方程，得到一个线性积分方程。这种方法的一个主要优点是它避免了求解由微分方程直接离散化而产生的稀疏但条件差的线性系统。相反，线性积分方程的离散化导致一个条件良好的密集线性系统，对于该系统，克雷洛夫子空间方法收敛速度很快。从历史上看，使用积分方程迭代方法的缺点是每一步都需要计算密集的矩阵-向量乘法。然而，随着快速多极方法（FMM）的出现，这些可以在最优时间内进行评估。然而，为了使该方法具有鲁棒性和实用性，还存在几个问题。特别是，对于材料性质具有强梯度的问题，有效的预处理策略尚未得到很好的发展。对模型问题的一些多层方法的初步分析是有希望的。我们计划的研究项目将经典数学（偏微分方程、积分方程、势理论）与科学计算研究（快速求和、迭代求解、自适应网格细化、域分解）和重要应用（多孔介质和变密度流）相结合。这项工作的跨学科性质为培养研究生成为计算科学家提供了肥沃的土壤。在上述应用领域获得的经验将产生广泛的影响，因为所创造的技术可以转移到环境、地球物理、生物和工程科学的许多其他应用领域。

The focus of this proposal is on the mathematical analysis and efficient implementation of integral equation based methods (IEM) for the solution of physically important, spatially inhomogeneous or "variable coefficient" elliptic equations. These arise in applications which require the modeling of complex fluid flows, such as combustion or ground water pollution. They also arise in the design of novel fluidic Micro-Electro-Mechanical Systems (MEMS), as well as solid-state semiconductor devices. Over the last decade, IEMs have proven to be very successful and are now widely used in electromagnetic, elastic, and fluid dynamic modeling when the governing equation is of "constant coefficient" type. In order to address more general, variable coefficient problems, we represent the solution as a volume integral using the Green's function for a simple, nearby problem convolved with an unknown density. Inserting this representation into the original elliptic equation leads to a linear integral equation. A principal advantage of this approach is that it avoids solving the sparse but poorly conditioned linear systems that result from direct discretization of the differential equation. Discretization of the linear integral equation leads, instead, to a well conditioned dense linear system, for which Krylov subspace methods converge quickly. Historically, the disadvantage of using iterative methods with integral equations was the cost of computing the dense matrix-vector multiplications which are needed at every step. With the advent of the fast multipole method (FMM), however, these can be evaluated in optimal time. Nevertheless, in order for the method to be robust and practical, several issues remain. In particular, effective preconditioning strategies for problems with strong gradients in the material properties are not well developed. Preliminary analysis of some multilevel approaches for model problems are promising.Our planned research program combines classical mathematics (PDEs, integral equations, potential theory) with scientific computing research (fast summation, iterative solvers, adaptive mesh refinement, domain decomposition) and important applications (porous media and variable density flows). The interdisciplinary nature of the work provides fertile ground for training graduate students as computational scientists. Experience gained in the application areas mentioned will have broad impact, since the techniques created can be transferred to many other application areas in the environmental, geophysical, biological, and engineering sciences.