Parallel Multilevel PDE Solvers on Unstructured Meshes

本项目将研究适用于分布式和共享内存并行计算架构的高效多网格（MG）和域分解（DD）算法，以解决任意非结构化网格上的椭圆和非椭圆问题。本文将重点研究DD和MG的三个方面：(1)各种DD和MG算法在并行计算机上的性能，特别关注通信和缓存延迟。(2)针对椭圆型和平流型主导问题，设计优化的DD和MG“灰盒”库，利用用户提供的可选信息。(3)适用于平流主导问题的DD和MG算法的理论分析和实际设计。将特别强调DD和MG算法，用于解决由各种大规模科学计算问题引起的离散化矩阵，例如用于平流主导问题的计算流体动力学（CFD），图像处理，光刻以及微芯片性能和制造的建模。这些实际问题的非椭圆行为使得已知的多重网格和分解理论不充分，并有助于激发由算法开发，理论分析和实际应用组成的平衡努力。

This project will investigate efficient multigrid (MG) and domain decomposition (DD) algorithms for elliptic and non-elliptic problems on arbitrary unstructured meshes, which are suitable for distributed and shared memory parallel computing architectures. Three aspects of DD and MG will be emphasized in this work: (1) Performance of various DD and MG algorithms on parallel computers with particular attention paid to communication and cache memory latency. (2) Design of optimized DD and MG "gray box" libraries for elliptic and advection dominated problems, which exploit optional information supplied by the user. (3) Theoretical analysis and practical design of DD and MG algorithm appropriate for advection dominated problems. Particular emphasis will be placed on DD and MG algorithms for solving the discretization matrices arising from a variety of large scale scientific computing problems, such as computational fluid dynamics (CFD) for advection dominated problems, image processing, photolithography, and the modeling of microchip performance and fabrication. The non-elliptic behavior of these practical problems renders the known multigrid and decomposition theory inadequate and serves to motivate a balanced effort consisting of algorithmic development, theoretical analysis, and practical application.