Novel Finite Element Methods for Three-Dimensional Anisotropic Singular Problems

椭圆型偏微分方程在各种实际模型中都具有奇异解。数值模拟的效果很大程度上依赖于光滑性，而在存在奇异点时，数值模拟的效果会严重下降。求解奇异偏微分方程的有效有限元方法（fem）的发展一直是计算数学的中心焦点；然而，现有的方法大多是针对二维问题的。有效的三维（3D）方法的设计涉及更多的技术，但探索较少，主要是由于三维几何和奇点各向异性结构的限制。现有的三维算法通常是基于四面体的，对区域几何形状和多项式度敏感，这限制了它们在实际计算中的应用。本项目旨在开发一种新的网格算法，该算法结构良好，可以灵活地处理不同的3D元素，并且实现简单。新算法有望显著提高各向异性解频繁出现的领域的三维数值模拟的有效性，包括航空航天工程（如飞机设计）、机械工程（如民用基础设施中的裂纹扩展）、流体力学和电磁学中的数学模型。这个研究项目有两个主要组成部分。(1)创新数值算法。研究人员计划开发一种新的各向异性网格，以促进最优fem近似三维各向异性奇异解。网格结构遵循简单，明确和统一的方法，并适用于四个基本的3D元素：四面体，六面体，楔形和金字塔。这些算法具有非常规但易于实现的特性，在早期的数值测试中显示出有效性。(2)严谨的理论研究与应用。研究者计划设计新的分析工具来证明和扩大新的fem的应用，特别是当网格是各向异性的。这包括(i)最优误差分析；（ii）三维各向异性问题的新规则估计；（iii）基于各向异性网格的线性系统的快速数值求解器；（iv）在高性能计算环境中的高效实现。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Elliptic partial differential equations can possess singular solutions in various practical models. The efficacy of numerical simulation highly depends on smoothness and can severely deteriorate in the presence of singularities. The development of effective finite element methods (FEMs) for singular partial differential equations has been a central focus in computational mathematics; however, most of the established methods are for two-dimensional problems. The design of effective three-dimensional (3D) methods is more technically involved and less explored, largely due to the constraints imposed by the 3D geometry and by anisotropic structure in singularities. Existing 3D algorithms are usually tetrahedron-based and sensitive to the domain geometry and to the degree of polynomials, which limits their applications in practical computing. This project aims to develop a novel mesh algorithm that is well-structured, flexible with different 3D elements, and simple in implementation. The new algorithm is expected to significantly improve the effectiveness of 3D numerical simulations in areas where anisotropic solutions frequently occur, including mathematical models in aerospace engineering (e.g., aircraft design), in mechanical engineering (e.g., crack propagation in civil infrastructure), in fluid mechanics, and in electromagnetism.This research project has two main components. (1) Innovative numerical algorithms. The investigator plans to develop a new family of anisotropic meshes that facilitate optimal FEMs approximating 3D anisotropic singular solutions. The mesh construction follows a simple, explicit, and unified approach and applies to four basic 3D elements: the tetrahedron, hexahedron, wedge, and pyramid. With unconventional but implementation-friendly features, these algorithms have shown effectiveness in early numerical tests. (2) Rigorous theoretical investigations and applications. The investigator plans to devise new analytical tools to justify and broaden the applications of the new FEMs, especially when the mesh is anisotropic. This includes (i) optimal error analysis; (ii) new regularity estimates for 3D anisotropic problems; (iii) fast numerical solvers for linear systems from anisotropic meshes; and (iv) efficient implementations in high-performance computing environments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.