基于 PDE 特性的特征值计算新型计算模式研究

目前 PDE 特征值问题的传统计算模式可描述为：粗网格生成、通过有限元建立离散格式并形成刚度矩阵与质量矩阵、调用代数特征值软件包、网格加密、迭代计算直至收敛等过程，其核心是把 PDE 特征值计算转化为一系列由网格加密所形成的离散系统代数特征值序列的极限，该模式方便通用，计算复杂性高，特别对于大型应用问题，迭代收敛往往慢得不可接受。. 我们认为，传统计算模式的主要瓶颈在于，它未能充分利用 PDE 特性与代数特征值计算之间的强耦合关系。本申请项目面向三类典型应用问题：信息领域的形状识别、科学计算领域的基于特征值分布的预条件子和物理领域的电子结构的能带计算，充分考虑 PDE 问题的几何特性、分析特性、离散特性、基底特性等，探索一条基于PDE特性、区别于目前国内外基于网格细分的传统算法的特征值问题新型计算模式。

At present，the mainstream modeling of PDE eigenpairs computation is described as following: coarse mesh generation, finie element discretization to obtain the stfiiness matrix and mass matrix on the coarse meshes, calling algebraic eigenpairs computation packages to get the needed approximations, h-version mesh refinement, iteration computation to improve their accuration until the tolerance is satisfied. . The essence of such traditional modeling is that, it transforms the PDE eigenpair computation to a series of algebraic eigenvalues approximation by means of h-refinement of discretization meshes. Though it is convenient and popular, the computation complexity is relatively high, and it may lead to too many iteration process for large-scale PDE application problems.. The bottleneck of traditional modeling is that, it ignores the strong correlation between PDE feature and algebraic eigenvalue problems. Based on backgrounds of three important application problems ( including shape-DNA recognition of information science, eigen-based preconditioner of scientific computing and energy calculation for electronic structure of optical physics ), we research some new eigenpair computation algorithms for PDE problem in order to make fully use of the geometric properties、analytic properties、discretization patterns and the properties of basic shape functions. Our computation modeling for PDE eigenpair problem will dramatically decrease the computation complexity comparing with the current traditional modeling.