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.
