Theory and Implementation of Novel Numerical Methods for Equations with Singularities

椭圆型偏微分方程组是各种科学学科中的基本数学模型，其解的逼近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逼近的效果。针对科学计算中的主要问题，奇异解的有限元方法的研究已经产生了许多有效的算法，但这些算法大多是针对二维奇异问题的。三维奇异解的有限元逼近这一领域的探索要少得多，也更具挑战性。由于奇异点的各向异性多尺度特性和三维几何的复杂性，现有的方法复杂且难以实现，并且还缺少一些关键的理论分析。该项目将大大提高现有数值模拟在许多领域的有效性，这些领域的多维计算是必不可少的。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Black-Scholes模型，流体和电磁场的建模，以及量子力学中薛定谔方程的计算。在这个项目中，PI提出了对椭圆型偏微分方程奇异解的有限元方法(FEMS)的系统研究，特别是在三维情况下。针对基本的理论和数值问题，这项研究包括两个主要部分。(I)创新的数值进展：开发新的三维网格划分算法。这些网格结构简单、显式且结构良好，能够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)严格的理论研究：(1)新函数空间的正则性估计；(2)二维梯度网格和所提出的三维网格在能量和非能量范数上的尖锐误差分析；(3)基于多重网格的快速数值求解器；(4)对所提出的三维网格的后验估计以及对其他具有奇异性的三维偏微分方程组的推广。本文的研究推进了三维奇异解有限元分析的前沿，为三维偏微分方程组数值算法的发展提供了新的思路，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最优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.