Integral Equation Methods for Variable Coefficient Elliptic Problems and Applications

这一建议的重点是基于积分方程式的方法(IEM)的数学分析和有效实施，以求解物理上重要的、空间不均匀的或“变系数”的椭圆型方程。这些问题出现在需要对复杂的流体流动进行建模的应用中，例如燃烧或地下水污染。它们也出现在新型射流微电子机械系统(MEMS)以及固态半导体器件的设计中。在过去的十年中，IEMS已被证明是非常成功的，现在被广泛应用于电磁、弹性和流体动力学建模中，当控制方程为“常系数”类型时。为了处理更一般的变系数问题，我们使用格林函数将解表示为一个简单的、附近的与未知密度卷积的问题的体积积分。将此表示法插入原始椭圆型方程即可得到一个线性积分方程式。这种方法的一个主要优点是它避免了求解稀疏但条件不佳的线性系统，这些线性系统是由微分方程的直接离散化引起的。取而代之的是，线性积分方程的离散化导致了一个条件良好的稠密线性系统，Krylov子空间方法对该系统收敛很快。在历史上，使用积分方程迭代法的缺点是每一步都需要计算密集的矩阵-向量乘法的成本。然而，随着快速多极子方法(FMM)的出现，这些方法可以在最佳时间进行评估。然而，为了使该方法稳健和实用，仍然存在几个问题。特别是，对于材料性质具有强梯度的问题，有效的预适应策略还没有很好地开发出来。我们计划的研究计划将经典数学(偏微分方程组、积分方程组、位势理论)与科学计算研究(快速求和、迭代求解器、自适应网格细化、区域分解)和重要应用(多孔介质和变密度流动)相结合。这项工作的跨学科性质为将研究生培养成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.