Multilevel Schwarz Preconditioners for Adaptive High-Order Discontinuous Galerkin Methods

自适应高阶间断伽辽金方法的多级 Schwarz 预处理器

• 批准号: 0612448
• 负责人: Luke Olson
• 金额: --
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0612448&HistoricalAwards=false
• 关键词: Multilevel; Schwarz; Preconditioners; Adaptive; Order

Problems that exhibit wave-like behavior, such as the Helmholtz equation and Maxwell's equation, often benefit immensely from the use of high-order discretization schemes. However, as the resolution of the desired numerical approximation increases, so does the computational complexity in solving the resulting system of equations. A multilevel solution technique is needed to achieve a (nearly) scalable process. A number of factors influence the efficiency of a multilevel method, most prominently the grid structure and underlying discretization scheme. Preconditioned Schwarz-based methods have previously shown to handle a wide range of problems. Most notably, high-order methods for positive definite problems, unstructured and non-nested grids, and indefinite operators. The proposed research will study Schwarz-based preconditioning for hp adaptive discontinuous spectral element discretizations of the indefinite time-harmonic Maxwell's equation. The research will focus on the discontinuous Galerkin method, which enables straightforward adaptivity, and will utilize the previous success of domain decomposition methods to extend recent results for the indefinite Helmholtz equation on similar grids.Electromagnetic principles find application in a broad range of industries. Consumer products such as cell phones, antenna development in the military, and many energy applications at the national laboratories rely on the fundamentals of electromagnetics. Computational simulation of this physical behavior is increasingly popular from an environmental, financial, and practical standpoint. Still, due to the underlying nature of these physical laws, the simulation process on large supercomputers is not yet efficient in a number of cases. The goal of this project is to further develop the solution techniques in the simulation process in an effort to improve efficiency. More efficient numerical algorithms lead to the ability to solve larger problems with higher accuracy, offering scientists a better view of the physical model. Specifically, the proposed research attacks the problem in a divide-and-conquer approach in a number of different respects. First, the mathematical laws of electromagnetics (Maxwell's equations) are approximated in a way that reduces unnecessary computational cost, called adaptive spectral elements. The challenge is that often this approach leads to an increasingly difficult second step by forming a large matrix of dependencies. This second step is the focus of the proposed work whereby the problem is decoupled by a process called domain decomposition. The research concentrates on improving the robustness and computational scalability.

表现出波动行为的问题，如亥姆霍兹方程和麦克斯韦方程，通常从高阶离散格式的使用中受益匪浅。 然而，随着所需数值近似的分辨率增加，求解所得方程组的计算复杂性也增加。 需要一种多级解决方案技术来实现（几乎）可扩展的过程。 许多因素影响多级方法的效率，最突出的是网格结构和底层离散方案。 预处理施瓦茨为基础的方法以前已经证明，以处理广泛的问题。 最值得注意的是，高阶方法的正定问题，非结构化和非嵌套网格，和不定算子。 本研究将研究基于Schwarz预处理的不定时谐麦克斯韦方程的hp自适应间断谱元离散。研究将集中在不连续Galerkin方法，它可以直接自适应，并将利用区域分解方法以前的成功，以扩展最近的结果为不定Helmholtz方程在类似的grids.Electromagnetic原则找到应用在广泛的行业。 消费类产品，如手机，天线在军事上的发展，并在国家实验室的许多能源应用依赖于电磁学的基本原理。 从环境、财务和实践的角度来看，这种物理行为的计算模拟越来越受欢迎。 尽管如此，由于这些物理定律的基本性质，在大型超级计算机上的模拟过程在许多情况下还不是有效的。 该项目的目标是进一步开发模拟过程中的解决方案技术，以提高效率。 更有效的数值算法能够以更高的精度解决更大的问题，为科学家提供更好的物理模型视图。 具体而言，所提出的研究攻击的问题，在一个分而治之的方法在许多不同的方面。 首先，电磁学的数学定律（麦克斯韦方程）以减少不必要的计算成本的方式近似，称为自适应谱元素。 挑战在于，这种方法通常会导致越来越困难的第二步，形成一个大型的依赖矩阵。 这第二步是重点提出的工作，即问题是解耦的过程称为域分解。 研究集中在提高鲁棒性和计算的可扩展性。