偏微分方程组的代数多层迭代法

提高线性解法器算法的适用性、收敛性、稳健性和扩展性是大规模科学计算的核心问题之一，而当前计算硬件体系结构的复杂程度使之变得更加重要。本项目针对几类偏微分方程组（如Stokes方程、Darcy方程、油藏模拟、半导体模拟）的离散线性系统，研究纯代数和半代数多层迭代法的算法设计、分析和实现。多个物理量的耦合效应、非结构网格或自适应网格所带来的不规则稀疏结构、现代计算机体系结构（主要是异构并行结构）的复杂性等给线性求解器算法发展带来了巨大挑战。为了有效地应对这些挑战，本项目计划：为几类偏微分方程组的离散线性系统量身定制基于空间位置和基于物理变量的初始化方法；基于精确及非精确两层网格法的收敛性理论，设计稀疏的理想插值算子和有效的粗空间求解方法；讨论耦合方程组的解耦方法和预条件方法之间的相互作用；利用非结构网格的拓扑信息和辅助空间方法获取粗空间及磨光算子，提高求解速度和并行可扩展性。

Many numerical simulation applications require solving large linear algebraic equations arising from systems of partial differential equations (PDEs) and the linear solver is usually the dominating bottle-neck in the whole simulation. In this sense, improving applicability, convergence, robustness, and scalability of linear solution methods plays a key role in these applications. Recent developments in heterogeneous computer architectures make linear solution methods more important in practice. In this project, we will focus on designing, analysis, and implementation of pure-algebraic and semi-algebraic multilevel iterative solvers for linear systems arising from discretizations of PDE systems like the Stokes equation, the Darcy’s equation, the black-oil equation in petroleum reservoir, and the drift-diffusion equation in semiconductor simulation. The main difficulties of this proposed project include, but not limited to, coupling of multiple physical unknowns with different physical properties, irregular sparse pattern due to unstructured or locally refined grids, and complexity of modern (heterogeneous) computer architectures. In order to deal with these challenges, we will design point-based and unknown-based setup methods for multilevel iterative solvers; we will construct sparse prolongation operators and effective coarse-level solvers based on theories of exact and inexact two-level methods; we will discuss relations between decoupling and preconditioning methods; we will enhance convergence rate and parallel scalability of multilevel iterative solvers by making use of accessible grid topology and auxiliary space methods.