Multilevel Schwarz Preconditioners for Adaptive High-Order Discontinuous Galerkin Methods

表现出波动行为的问题，如亥姆霍兹方程和麦克斯韦方程，通常从使用高阶离散化方案中受益匪浅。然而，随着期望数值近似的分辨率增加，求解结果方程组的计算复杂度也随之增加。需要一种多层解决方案技术来实现（几乎）可伸缩的流程。许多因素影响多层方法的效率，最突出的是网格结构和底层离散化方案。预先条件的基于schwarz的方法已经被证明可以处理各种各样的问题。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。提出的研究将研究基于schwarz的预处理对不确定时调和麦克斯韦方程的hp自适应不连续谱元离散化。研究将集中在不连续伽辽金方法上，该方法具有直接的自适应性，并将利用先前成功的区域分解方法来扩展类似网格上不定亥姆霍兹方程的最新结果。电磁原理在各行各业都有广泛的应用。诸如手机、军用天线开发以及国家实验室的许多能源应用等消费产品都依赖于电磁学的基础知识。从环境、金融和实用的角度来看，这种物理行为的计算模拟越来越受欢迎。然而，由于这些物理定律的基本性质，在许多情况下，大型超级计算机上的模拟过程还不是很有效。该项目的目标是进一步开发仿真过程中的解决方案技术，以提高效率。更高效的数值算法能够以更高的精度解决更大的问题，为科学家提供更好的物理模型视图。具体来说，拟议的研究在许多不同方面以分而治之的方法来解决这个问题。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近似，称为自适应谱元素。挑战在于，这种方法通常会导致第二步变得越来越困难，因为它形成了一个很大的依赖矩阵。第二步是建议工作的重点，通过称为域分解的过程将问题解耦。研究的重点是提高鲁棒性和计算可扩展性。

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.