Theory and Implementation of Novel Numerical Methods for Equations with Singularities

椭圆型偏微分方程是各种科学学科中必不可少的数学模型，而这些方程的近似解的方法的发展一直是计算数学的中心焦点。数值方法的性能一般取决于所研究的解的光滑性。在实际应用中，解中的奇异性会严重影响数值近似的有效性。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。由于奇异点的各向异性多尺度特性和三维几何的复杂性，现有的方法既复杂又难以实现，而且还缺少一些关键的理论分析。该项目将显著提高许多需要多维计算的领域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、医学成像中的弹性学、金融中的布莱克-斯科尔斯模型、流体和电磁场的建模，以及量子力学中Schrödinger方程的计算。在本项目中，PI对椭圆偏微分方程奇异解的有限元方法（fem）进行了系统的研究，特别是在三维中。针对基础理论和数值问题，本研究有两个主要组成部分。(1)数值方面的创新进展：新的三维网格划分算法的发展。这些网格简单、明确、结构良好，可以有效地捕捉奇异解的局部行为，从而得到最优的有限元模型。(2)严谨的理论研究：(1)在新的函数空间中有明显的正则性估计；(2)对二维梯度网格和建议的三维网格的能量和非能量范数进行了尖锐误差分析；(3)基于多网格的快速数值求解；(4)对提出的三维网格进行后验估计，并扩展到其他具有奇异点的三维偏微分方程。本文的研究将进一步推动三维奇异解有限元分析的前沿，为三维偏微分方程的新型数值算法的发展带来新的思路，具有更广泛的应用前景。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。

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.