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

具有低正则性数据的椭圆型偏微分方程（PDE）（即，奇异解和低正则性系数）经常出现在来自不同科学学科的数学模型中。这些方程在一般情况下提出了多尺度的字符，这增加了问题的复杂性，并提出了许多挑战的有限元逼近和多重网格格式的设计。尽管计算界不断发展，但一些基本问题仍然悬而未决。解决理论分析和实际实施中的主要问题，该建议的目的是系统地调查的各个方面的有限元法（FEM）和多重网格（MG）方法解决椭圆偏微分方程与低正则性数据，范围广泛的偏微分方程的理论估计的发展国家的最先进的数值算法。具体而言，本文的研究主要包括两个方面：（1）奇异解的最优有限元方法，包括（1）建立一个统一的框架来分析加权Sobolev空间中的各种奇异解;（2）发展和实现有效的分级算法来提高逼近这些奇异解的数值解的精度; 2）轴对称问题的MG理论，包括轴对称拉普拉斯算子的MG基本圈的估计和加权空间中轴对称Stokes问题的新smoothers的设计。拟议的研究将产生新的先验结果（例如，（适定性和正则性），充分利用梯度网格技术求解奇异解的潜力，扩展MG理论在轴对称方程上的应用范围，并在求解偏微分方程方面培养创新思想，使其具有更广泛的应用。所提出的研究在科学和工程的各个领域有许多应用。椭圆型偏微分方程的一类奇异解来源于传输问题中计算区域的非光滑性和界面的非光滑性。对这些奇异性的研究将产生有效的数值算法，解决航空航天工程（飞机设计），机械工程（裂纹扩展）和医学成像弹性成像（模拟人体组织中不同程度的刚度）中的问题。从奇异系数出发研究奇异解，将为量子力学中具有各种奇异势的Schroedinger方程的求解提供新的理论结果和现代有限元技术。此外，轴对称模型的MG分析将导致一个更完整的MG理论在奇异空间，反过来又带来了新的快速数值求解器，这些方程，经常使用的流体和电磁场。这些领域对国家安全、新能源开发和新型医学研究具有深远影响。该项目的成果将通过与其他学者合作、发表同行评审文章以及在专业会议上发表演讲等方式传播。随着软件包的开发，PI还希望为数学和工程专业的高年级本科生/研究生设计一个以项目为导向的有限元法课程，这将使学生更好地理解算法和宝贵的编程经验。

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.