Novel Finite Element Methods for Three-Dimensional Anisotropic Singular Problems

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Elliptic partial differential equations can possess singular solutions in various practical models. The efficacy of numerical simulation highly depends on smoothness and can severely deteriorate in the presence of singularities. The development of effective finite element methods (FEMs) for singular partial differential equations has been a central focus in computational mathematics; however, most of the established methods are for two-dimensional problems. The design of effective three-dimensional (3D) methods is more technically involved and less explored, largely due to the constraints imposed by the 3D geometry and by anisotropic structure in singularities. Existing 3D algorithms are usually tetrahedron-based and sensitive to the domain geometry and to the degree of polynomials, which limits their applications in practical computing. This project aims to develop a novel mesh algorithm that is well-structured, flexible with different 3D elements, and simple in implementation. The new algorithm is expected to significantly improve the effectiveness of 3D numerical simulations in areas where anisotropic solutions frequently occur, including mathematical models in aerospace engineering (e.g., aircraft design), in mechanical engineering (e.g., crack propagation in civil infrastructure), in fluid mechanics, and in electromagnetism.This research project has two main components. (1) Innovative numerical algorithms. The investigator plans to develop a new family of anisotropic meshes that facilitate optimal FEMs approximating 3D anisotropic singular solutions. The mesh construction follows a simple, explicit, and unified approach and applies to four basic 3D elements: the tetrahedron, hexahedron, wedge, and pyramid. With unconventional but implementation-friendly features, these algorithms have shown effectiveness in early numerical tests. (2) Rigorous theoretical investigations and applications. The investigator plans to devise new analytical tools to justify and broaden the applications of the new FEMs, especially when the mesh is anisotropic. This includes (i) optimal error analysis; (ii) new regularity estimates for 3D anisotropic problems; (iii) fast numerical solvers for linear systems from anisotropic meshes; and (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.