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

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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.