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

具有复杂边界条件的偏微分方程是跨学科的基本模型。通过CBCs，我们的意思是边界条件（BC）比基本的Dirichlet或Neumann BC更复杂，具有通常用于通用数值算法的说明和理论研究的规则边界数据。这些CBCs往往会导致不同类型的奇异解，严重恶化的数值逼近的功效。该项目将为重要应用中出现的CBCs问题开发简单，高效和强大的数值方法。例如，在结构力学中，低规则性边界数据（例如，不连续性或分布）用于模拟作用在边界上的载荷或集中力的突然变化; Robin BC与Dirichlet BC组合用于模拟在完整电极模型中、在奇摄动辐射问题中以及在将量子结构嵌入宏观流中时发生的阻抗BC; Ventcel BC用于模拟热传导过程;而包含高阶微分算子的CBCs对于双调和方程模拟薄板的静态载荷是必不可少的。还注意到，不同的CBC对于血液动力学应用中的流体动力学、电磁场和流体-结构相互作用中的模型是重要的。此外，PI希望该项目的教育部分将展示科学计算领域令人兴奋的创新，并鼓励来自不同背景的未来劳动力在STEM领域接受教育。该研究项目是关于正则性分析和有限元方法（FEM）的开发，用于解决具有CBCs的二阶和四阶椭圆（PDE）。对于二阶PDE，CBC包括低规则性边界数据和各种BC（例如，Dirichlet，Neumann，mixed，Robin，and Ventcel）.对于四阶偏微分方程，所考虑的CBCs是经典的BC，特别是与双调和算子。这些CBCs，加上域的几何形状，在实践中产生一些最常见的解决方案的奇异性。解决关键的分析和计算问题，这项研究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），高效（有效的数值逼近）和鲁棒性（适用于一般多边形或多面体域）的各种奇异解决方案，由于CBCs的FEM。(II)严谨的理论研究和应用。PI将设计新的分析工具，以证明和扩大拟议FEM的应用。这包括（i）CBCs问题的新适定性和规律性估计;（ii）最优误差分析;（iii）对3D和其他实用模型的扩展;（iv）高性能计算环境中的高效实现。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.