Theory and Implementation of Novel Numerical Methods for Equations with Singularities

奇异性方程新数值方法的理论与实现

• 批准号: 1418853
• 负责人: Hengguang Li
• 金额: $ 15.28万
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1418853&HistoricalAwards=false
• 关键词: Theory; Implementation; Novel; Numerical; Methods

Elliptic partial differential equations are essential mathematical models in various scientific disciplines, and the development of methods for approximating solutions of these equations has been a central focus of computational mathematics. The performance of numerical methods in general depends on the smoothness of the solutions under study. Quite common in practical applications, singularities in the solution can severely deteriorate the efficacy of the numerical approximation. Addressing major concerns in scientific computations, the study of finite element methods for singular solutions has led to many effective algorithms, but most of them are for two-dimensional singular problems. The area of finite element approximations for three-dimensional singular solutions is much less explored and much more challenging. Due to the anisotropic multiscale character of the singularity and the complexity of three-dimensional geometry, the existing methods are complicated and difficult to implement and are still missing some critical pieces of theoretical analysis. This project will significantly improve the effectiveness of existing numerical simulations in many areas where multi-dimensional computations are essential. These areas include aircraft design in aerospace engineering, crack propagation in mechanical engineering, elastography in medical imaging, Black-Scholes models in finance, modeling of fluids and of electromagnetic fields, and computation for the Schrödinger equation in quantum mechanics. In this project, the PI proposes a systematic research on finite element methods (FEMs) for singular solutions of elliptic PDEs, especially in 3D. Targeting fundamental theoretical and numerical issues, this research has two main components. (I) Innovative numerical advancements: the development of new 3D meshing algorithms. Simple, explicit, and well structured, these meshes can effectively capture the local behavior of the singular solution and lead to optimal FEMs. (II) Rigorous theoretical investigations: (1) sharp regularity estimates in new function spaces; (2) sharp error analysis in energy and non-energy norms on both 2D graded meshes and the proposed 3D meshes; (3) fast multigrid-based numerical solvers on these meshes; (4) a-posteriori estimates on the proposed 3D meshes and extensions to other 3D PDEs with singularities. Pushing forward the frontier of the FEMs for 3D singular solutions, this proposed research will bring new ideas to the development of novel numerical algorithms for 3D PDEs with broader applications. In addition, the sharp non-energy error estimates and a-posteriori analysis can provide theoretical justifications for nonlinear models and optimization control problems.

椭圆偏微分方程是各个科学学科中必不可少的数学模型，并且开发这些方程的近似解的方法一直是计算数学的中心焦点。数值方法的性能通常取决于所研究的解的平滑度。在实际应用中很常见，解中的奇点会严重降低数值近似的效果。针对科学计算中的主要问题，对奇异解的有限元方法的研究已经产生了许多有效的算法，但其中大多数是针对二维奇异问题的。三维奇异解的有限元近似领域的探索要少得多，而且更具挑战性。由于奇点的各向异性多尺度特征和三维几何的复杂性，现有方法复杂且难以实现，并且仍然缺少一些关键的理论分析。该项目将显着提高多维计算必不可少的许多领域中现有数值模拟的有效性。这些领域包括航空航天工程中的飞机设计、机械工程中的裂纹扩展、医学成像中的弹性成像、金融中的布莱克-斯科尔斯模型、流体和电磁场建模以及量子力学中薛定谔方程的计算。在该项目中，PI 提议对椭圆偏微分方程奇异解的有限元方法 (FEM) 进行系统研究，特别是在 3D 中。针对基础理论和数值问题，本研究有两个主要组成部分。 (I) 创新的数值进展：新的 3D 网格划分算法的开发。这些网格简单、明确且结构良好，可以有效捕获奇异解的局部行为并产生最佳的有限元模型。 （二）严谨的理论研究：（1）新函数空间中尖锐的规律性估计； (2) 对 2D 分级网格和所提出的 3D 网格进行能量和非能量范数的锐误差分析； (3) 这些网格上基于多重网格的快速数值求解器； (4) 对所提出的 3D 网格的后验估计以及对其他具有奇点的 3D PDE 的扩展。这项研究推动了 3D 奇异解 FEM 的前沿发展，将为开发具有更广泛应用的 3D PDE 新型数值算法带来新的思路。此外，尖锐的非能量误差估计和后验分析可以为非线性模型和优化控制问题提供理论依据。