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.

大规模的非对称特征值问题在科学和工程计算的各个领域都很普遍。然而，它们的有效处理具有挑战性，并且对其解决方案算法的研究是有限的。在这项研究中，基于自适应细化技术和多级校正方案，设计了一种新的用于校正非对称特征值问题的多级校正自适应有限元方法。与经典的自适应有限元方法不同，经典的自适应有限元方法需要在每个自适应细化空间中解决一个非对称特征值问题，而我们的方法则需要解决当前精化空间中的一个对称线性边值问题和一个在富集校正空间中的小规模非对称特征值问题。 。由于直接在自适应空间中求解大型非对称特征值问题很费时，因此该方法在求解对称线性边界值问题时可以达到与经典自适应算法几乎相同的效率。另外，对相应的收敛性和最优复杂度进行了理论验证和数值验证。