研究期間：10108~10207;Many scientific and engineering applications require accurate and fast numerical solution of large scale sparse algebraic polynomial eigenvalue problems (PEVPs) arising from some appropriate disretization of partial differential equations. The polynomial Jacobi-Davidson (PJD) algorithm has been numerically shown as a promising approach for the PEVPs and has gained its popularity. The PJD algorithm is a subspace method, which extracts the candidate approximate eigenpair from a search space and the space undated by embedding the solution of the correction equation at the JD iteration. The first aim of the project is to develop, implement, and study the multilevel PJD algorithm for PEVPs with emphasis on the fluid-structure interaction. The proposed multilevel PJD algorithm is based on the Schwarz framework. The initial basis for the search space is constructed on the current level by using the solution of the same eigenvalue problem but defined on the previous coarser grid. On the other hand, a parallel efficient multilevel Schwarz preconditioner is designed for the correction equation to enhance the scalability of the PJD algorithm, which plays a crucial property in parallel computing for large-scale problem solved by using a large number of processors. The second aim of this project is to provide a user-friendly scientific software package for scientists and engineers to meet their needs for the numerical solution of large-scale PEVPs with real-world applications with little programming efforts and the developers to solve more complex system such as the EVP and PDEs problem coupling problem. To achieve this goal, before the package publicly released, we plan to conduct extensive numerical experiments to determine the optimal values for the algorithmic parameters involved in the PJD algorithm, which will be used the default value in the package and hopefully to show its superiority compared with the state-of-the-art eigenvalue packages in terms of robustness, efficiency, and accuracy. In addition, several new features, such as harmonic Ritz extraction and deflection, are planned to be added to increase the applicability of the package for a variety of applications.
