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.

椭圆形的部分微分方程是各种科学学科中的基本数学模型，并且开发用于近似这些方程解决方案的方法已成为计算数学的核心重点。数值方法的性能通常取决于所研究的解决方案的平滑度。在实际应用中很常见，解决方案中的奇异性会严重恶化数值近似的效率。解决了科学计算中的主要问题，对单数解的有限元方法的研究导致了许多有效的算法，但其中大多数是针对二维奇异问题。三维奇异解决方案的有限元近似区域的探索要少得多，而挑战更大。由于奇异性的各向异性多尺度和三维几何形状的复杂性，现有方法是复杂的且难以实施的，并且仍然缺少一些理论分析的一些关键部分。该项目将显着提高在多维计算至关重要的许多领域中现有数值模拟的有效性。这些领域包括航空工程中的飞机设计，机械工程中的裂纹传播，医学成像中的弹性学，金融中的黑色 - choles模型，流体建模和电磁场的建模以及量子力学中Schrödinger方程的计算。该项目，PI提出了针对椭圆PDE的单数解的最终元素方法（FEM）的系统研究，尤其是在3D中。针对基本理论和数值问题，这项研究具有两个主要组成部分。 （i）创新的数值进步：新的3D网格划分算法的开发。这些网格简单，显式且结构良好，可以有效地捕获单数解决方案的局部行为并导致最佳FEM。 （ii）严格的理论投资：（1）新功能空间中的尖锐规律性估计； （2）在2D级网格和拟议的3D网格上的能量和非能量规范的急剧误差分析； （3）这些网格上的快速基于多移民的数值求解器； （4）对拟议的3D网格和其他3D PDE的a-tosterii估计。这项拟议的研究推动了3D单数解决方案的FEM人的前沿，将为3D PDE的新型数值算法带来新的想法，并具有更广泛的应用。此外，尖锐的非能误差估计和A-Tosteriori分析可以为非线性模型和优化控制问题提供理论上的理由。