Analysis and Novel Finite Element Methods for Elliptic Equations with Complex Boundary Conditions

复杂边界条件椭圆方程的分析和新颖的有限元方法

• 批准号: 2208321
• 负责人: Hengguang Li
• 金额: $ 22.03万
• 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: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2208321&HistoricalAwards=false
• 关键词: Analysis; Novel; Finite; Element; Methods

Partial differential equations (PDEs) with complex boundary conditions (CBCs) are essential models across scientific disciplines. By CBCs, we mean boundary conditions (BCs) that are more complex than the basic Dirichlet or Neumann BC with regular boundary data that are usually adopted for the illustration and theoretical study of general-purpose numerical algorithms. These CBCs often lead to different types of singular solutions that severely deteriorate the efficacy of the numerical approximation. This project will develop simple, efficient, and robust numerical methods for problems with CBCs that appear in important applications. For example, in structural mechanics, low regularity boundary data (e.g., discontinuities or distributions) are used to model sudden changes of loads or concentrated forces acting on the boundary; the Robin BC, combined with the Dirichlet BC, is used to model the impedance BC that occurs in complete electrode models, in singularly perturbed radiation problems, and in embedding of quantum structures into a macroscopic flow; the Ventcel BCs are used to model heat conduction processes; and CBCs involving high-order differential operators are essential for biharmonic equations to model the static loading of a thin plate. It is also noted that different CBCs are important for models in fluid dynamics, electromagnetic fields, and fluid-structure interactions in hemodynamics applications. In addition, the PI expects that the project's educational component will demonstrate exciting innovations in scientific computing and encourage the future workforce from diverse backgrounds to pursue education in STEM fields.The research project is on regularity analysis and on the development of finite element methods (FEMs) solving 2nd-order and 4th-order elliptic (PDEs) with CBCs. For 2nd-order PDEs, the CBCs include low regularity boundary data and various BCs (e.g., Dirichlet, Neumann, mixed, Robin, and Ventcel). For 4th-order PDEs, the CBCs under consideration are classical BCs especially associated with the biharmonic operator. These CBCs, together with the domain geometry, give rise to some of the most common solution singularities in practice. Addressing key analytical and computational issues, this research has two main components. (I) Innovative numerical algorithms. The PI will develop FEMs that are simple (easy to implement), efficient (effective in numerical approximation), and robust (applicable to general polygonal or polyhedral domains) for various singular solutions due to CBCs. (II) Rigorous theoretical investigation and applications. The PI will devise new analytical tools to justify and broaden the applications of the proposed FEMs. This includes (i) new well-posedness and regularity estimates for problems with CBCs; (ii) optimal error analysis; (iii) extensions to 3D and other practical models; (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.

具有复杂边界条件（CBC）的部分微分方程（PDE）是科学学科的基本模型。通过CBC，我们是指比基本的Dirichlet或Neumann BC更复杂的边界条件（BCS），其常规边界数据通常用于通用数值算法的插图和理论研究。这些CBC通常会导致不同类型的奇异溶液，从而严重恶化了数值近似的功效。对于出现在重要应用程序中的CBC问题的问题，该项目将开发简单，高效且可靠的数值方法。例如，在结构力学中，低规律性边界数据（例如不连续性或分布）用于建模作用在边界上的负载或集中力的突然变化； Robin BC与Dirichlet BC结合使用，用于模拟在完整电极模型，奇异扰动的辐射问题以及将量子结构嵌入宏观流中的阻抗BC。 Ventcel BCS用于建模热传导过程。涉及高阶差分运算符的CBC对于二旋转方程来说至关重要，以建模薄板的静态载荷。还注意到，不同的CBC对于流体动力学，电磁场和血液动力学应用中的流体结构相互作用很重要。此外，PI期望该项目的教育组成部分将在科学计算中展示令人兴奋的创新，并鼓励来自不同背景的未来员工在STEM领域进行教育。该研究项目是针对规则性分析和与CBC一起解决2阶和4阶和4th-ford Elliptic（PDES）的有限元方法（FEMS）的开发。对于二阶PDE，CBC包括低规律性边界数据和各种BC（例如Dirichlet，Neumann，Mixed，Mixper，Robin和Ventcel）。对于四阶PDE，所考虑的CBC是经典的BCS，尤其与Biharmonic运营商有关。这些CBC与域的几何形状一起产生了实践中最常见的解决方案奇异性。解决关键的分析和计算问题，这项研究具有两个主要组成部分。 （i）创新的数值算法。 PI将开发简单（易于实现的），有效（在数值近似方面有效）和稳健（适用于一般多边形或多面体结构域）的FEM，适用于CBC引起的各种单数解决方案。 （ii）严格的理论研究和应用。 PI将设计新的分析工具，以证明和扩大拟议中的FEM的应用。这包括（i）CBC问题的新适应性和规律性估算； （ii）最佳误差分析； （iii）扩展到3D和其他实用模型； （iv）在高性能计算环境中的有效实施。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。