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

具有低规律性数据（即奇异解和低规律性系数）的椭圆偏微分方程 (PDE) 经常出现在不同科学学科的数学模型中。这些方程通常呈现出多尺度特征，这增加了问题的复杂性，并对有限元近似和多重网格方案的设计提出了许多挑战。尽管计算社区不断发展，但一些基本问题仍然悬而未决。该提案针对理论分析和实际实施中的主要问题，旨在系统地研究有限元法（FEM）和多重网格（MG）方法求解具有低规律性数据的椭圆偏微分方程的各个方面，范围从偏微分方程的理论估计到最先进数值算法的开发。具体来说，所提出的研究包括两个主要部分：1）奇异解的最优有限元法，包括（I）建立一个统一的框架，用于分析加权Sobolev空间中的各种奇异解；（II）开发和实施有效的分级算法，以提高逼近这些奇异解的数值解的准确性； 2）轴对称问题的MG理论，包括轴对称拉普拉斯算子基本MG循环的估计以及加权空间中轴对称Stokes问题的新型平滑器的设计。拟议的研究将为加权空间中的各种奇异解产生新的先验结果（例如，适定性和正则性），统一奇异解的分级网格技术的全部潜力，扩展轴对称方程的 MG 理论的范围，并培育具有更广泛应用的求解偏微分方程的创新思想。所提出的研究在科学和工程的各个领域都有许多应用。椭圆偏微分方程的一类奇异解来源于计算域的非光滑性和传输问题中界面的非光滑性。对这些奇点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（模拟人体组织不同硬度水平）中的问题。奇异系数奇异解的研究将为解决量子力学中各种奇异势的薛定谔方程提供新的理论成果和现代有限元技术。此外，轴对称模型的 MG 分析将在奇异空间中产生更完整的 MG 理论，进而为这些在流体和电磁场中经常使用的方程带来新的快速数值求解器。这些都是对国家安全、新能源开发、新医学研究等产生深远影响的领域。该项目的成果将通过与其他学者的合作、发表同行评审的文章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专业的高年级本科生/研究生设计一个面向项目的有限元方法课程，让学生对算法有更好的理解，并获得宝贵的编程经验。

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.