Parallel Multilevel PDE Solvers on Unstructured Meshes

本项目将研究任意非结构网格上椭圆和非椭圆问题的高效多重网格（MG）和区域分解（DD）算法，这些算法适用于分布式和共享内存并行计算架构。 本文将着重研究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)优化的DD和MG“灰箱”库的椭圆和平流为主的问题，利用用户提供的可选信息的设计。 (3)对适用于平流占优问题的DD和MG算法进行了理论分析和实际设计。 特别强调将放在DD和MG算法解决离散化矩阵所产生的各种大规模的科学计算问题，如计算流体动力学（CFD）的平流为主的问题，图像处理，光刻，和微芯片的性能和制造的建模。 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足，并有助于激发平衡的努力，包括算法开发，理论分析和实际应用。

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.