Multilevel Schwarz Preconditioners for Adaptive High-Order Discontinuous Galerkin Methods

像亥姆霍兹方程和麦克斯韦方程这样表现出波动行为的问题，往往从高阶离散化格式的使用中受益匪浅。然而，随着所需数值近似的分辨率的提高，求解所得到的方程组的计算复杂性也随之增加。需要多层解决方案技术来实现(几乎)可伸缩的流程。影响多层方法效率的因素有很多，最突出的是网格结构和底层离散化方案。基于施瓦茨的预条件方法此前已被证明可以处理广泛的问题。最值得注意的是，正定问题的高阶方法，非结构化和非嵌套网格，以及不定算子。这项研究将研究基于Schwarz的预条件用于不定时间调和Maxwell方程的hp自适应间断谱元离散。研究将集中在具有直接自适应性的不连续Galerkin方法上，并将利用先前区域分解方法的成功来推广最近关于相似网格上不定Helmholtz方程的结果。电磁原理在广泛的工业中得到应用。消费产品，如手机，军事天线的开发，以及国家实验室的许多能源应用，都依赖于电磁学的基本原理。从环境、财政和实用的角度来看，对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潜在性质，在大型超级计算机上的模拟过程在许多情况下还不是很有效。本项目的目标是进一步发展模拟过程中的求解技术，以努力提高效率。更高效的数值算法能够以更高的精度解决更大的问题，为科学家提供了更好的物理模型视图。具体地说，这项拟议的研究在许多不同的方面以分而治之的方式解决了这个问题。首先，电磁学的数学定律(麦克斯韦方程)以一种减少不必要的计算成本的方式进行近似，称为自适应谱元素。挑战在于，这种方法通常会形成一个大型依赖矩阵，从而导致第二步变得越来越困难。第二步是拟议工作的重点，通过一个称为域分解的过程来分离问题。研究的重点是提高算法的健壮性和计算可伸缩性。

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.