(untitled) (Advances in Applied Mathematics and Mechanics, 2025)
Solving eigenvalue problems is an important subject in scientific computing. Classical numerical methods, such as finite element methods and spectral methods, have been developed with great success in applications. However, these methods struggle in the case of complex geometries. The recently developed random feature method has demonstrated its superiority in solving partial differential equations (PDEs), especially for problems with complex geometries. In this work, we develop random feature methods for solving elliptic eigenvalue problems. Our contributions include (1) using tailored separation-of-variables random feature functions to approximate eigenfunctions, (2) employing collocation points in the strong formulation or quadrature scheme (exact integration) in the weak formulation to handle the PDEs and boundary conditions, and (3) solving the generalized eigenvalue problem in which the number of conditions equals the number of unknowns. Through a series of examples from one dimension to three dimensions, we demonstrate the high accuracy and robustness of the proposed methods with respect to geometric complexity. We have made our source code publicly available at https://github.com/lingyun-2024/ The-code-of-RFM-EEP.git.