Novel Numerical Methods for Partial Differential Equations with Low-regularity Data

低正则数据偏微分方程的新数值方法

• 批准号: 1115714
• 负责人: Hengguang Li
• 金额: $ 6.01万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115714&HistoricalAwards=false
• 关键词: Novel; Numerical; Methods; Partial; Differential

Elliptic partial differential equations (PDEs) with low-regularity data (i.e., singular solutions and low-regularity coefficients) frequently appear in mathematical models from diverse scientific disciplines. These equations in general present a multiscale character, which increases the complexity of the problem and poses numerous challenges on the finite element approximation and on the design of multigrid schemes. Despite continuous developments from the computational community, some fundamental questions still remain open. Addressing major issues in both theoretical analysis and practical implementation, this proposal aims at a systematic investigation on various aspects of the finite element method (FEM) and multigrid (MG) methods solving elliptic PDEs with low-regularity data, with a wide range from the theoretical estimates of PDEs to the development of state-of-the-art numerical algorithms. In particular, the proposed research consists of two major components: 1) the optimal FEMs for singular solutions, including (I) the establishment of a unified framework for the analysis of a wide variety of singular solutions in weighted Sobolev spaces and(II) the development and implementation of effective grading algorithms to improve the accuracy of the numerical solution approximating these singular solutions; 2) the MG theory for axisymmetric problems, including the estimation of basic MG cycles for the axisymmetric Laplace operator and the design of new smoothers for the axisymmetric Stokes problem in weighted spaces. The proposed research will produce new a priori results (e.g., the well-posedness and regularity) for various singular solutions in weighted spaces, unitize the full potential of graded meshing techniques for singular solutions, expand the scope of the MG theory on axisymmetric equations, and foster innovative ideas on solving PDEs with broader applications. The proposed research has many applications in various fileds of science and engineering. A class of singular solutions of elliptic PDEs are from the non-smoothness of the computational domain and the non-smoothness of the interface in transmission problems. The study on these singularities shall produce effective numerical algorithms solving problems in aerospace engineering (aircraft design), in mechanical engineering (crack propagation), and in elastography of medical imagining (modeling different levels of stiffness in human tissues). The research on singular solutions from the singular coefficients shall provide new theoretical results and modern finite element techniques to tackle Schroedinger equations with various singular potentials in quantum mechanics. In addition, the MG analysis on axisymmetric models shall lead to a more complete MG theory in singular spaces and in turn bring new fast numerical solvers for these equations that are frequently used in fluids and i n electromagnetic fields. These are the fields that have profound impact on national security, development of new energy, and novel medical research. Results from this project will be disseminated through collaboration with other scholars, publication of peer-reviewed articles, and presentations at professional meetings. With the development of a software package, the PI also expects to design a project-oriented course on the finite element method for senior undergraduate/graduate students in math and engineering, which will equip the students with a better understanding on the algorithm and a valuable programming experience.

来自不同科学学科的数学模型中，经常出现具有低规范性数据（即单数解和低规范系数）的椭圆形部分微分方程（PDE）。通常，这些方程式具有多尺寸特征，这增加了问题的复杂性，并在有限元近似和多机计划的设计上构成了许多挑战。尽管计算界的不断发展，但一些基本问题仍然保持开放。该提案旨在解决理论分析和实际实施中的主要问题，旨在对有限元方法（FEM）的各个方面（fem）和多机（MG）方法进行系统调查，以求解具有低规范性数据的椭圆形PDE，其范围很广，从PDE的理论估计到了最高的数字算法的发展。尤其是拟议的研究由两个主要组成部分组成：1）最佳的唯一解决方案的FEM，包括（i）建立一个统一的框架，用于分析加权Sobolev空间中多种单数解决方案，以及（ii）开发和实施有效分级算法，以提高这些数字解决方案的准确性，以提高这些数字近似近似近似的近似signular sissimal近似近似近似近似近似signular signultimal signallimal ofdimoltim近似近似近似近似近似近似的溶液； 2）轴对称问题的MG理论，包括轴对称拉式算子的基本MG循环的估计以及在加权空间中为轴对称的Stokes问题设计的新SmoOther。拟议的研究将为加权空间中各种单数解决方案产生新的先验结果（例如，适当的和规律性），将分级网格划分技术的全部潜力定为单数解决方案的全部潜力，扩大了MG理论在Axisymmetric方程上的范围，并促进了解决广泛应用程序的创新思想。拟议的研究在科学和工程的各种申请中都有许多应用。一类椭圆形PDE的单数解来自计算域的非平滑度和在传输问题中界面的不平滑度。对这些奇异性的研究应产生有效的数值算法解决航空工程（飞机设计），机械工程（裂纹传播）和医学构想的弹性（建模人体组织中不同水平的刚度）中的有效数值算法。来自单数系数的奇异解决方案的研究应提供新的理论结果和现代有限元技术，以解决具有量子力学中各种奇异电位的Schroedinger方程。此外，轴对称模型的MG分析应导致更完整的MG理论在奇异空间中，进而为这些方程式带来新的快速数值求解器，这些方程经常用于流体和电磁场。这些领域对国家安全，新能源的发展和新颖的医学研究产生了深远的影响。该项目的结果将通过与其他学者的合作，同行评审文章的出版以及在专业会议上的演讲来传播。随着软件包的开发，PI还希望为数学和工程学领域的高级本科/研究生设计一个面向项目的课程，这将使学生对算法有更好的了解和有价值的编程体验。