Multilevel correction adaptive finite element method for solving nonsymmetric eigenvalue problems

Large-scale nonsymmetric eigenvalue problems are common in various fields of science and engineering computing. However, their efficient handling is challenging, and research on their solution algorithms is limited. In this study, a new multilevel correction adaptive finite element method is designed for solving nonsymmetric eigenvalue problems based on the adaptive refinement technique and multilevel correction scheme. Different from the classical adaptive finite element method, which requires solving a nonsymmetric eigenvalue problem in each adaptive refinement space, our approach requires solving a symmetric linear boundary value problem in the current refined space and a small-scale nonsymmetric eigenvalue problem in an enriched correction space. Since it is time-consuming to solve a large-scale nonsymmetric eigenvalue problem directly in adaptive spaces, the proposed method can achieve nearly the same efficiency as the classical adaptive algorithm when solving the symmetric linear boundary value problem. In addition, the corresponding convergence and optimal complexity are verified theoretically and demonstrated numerically.