許多科學和工程應用，需要精確與快速地對大型稀疏多項式特徵值問題求數值解。多項式Jacobi-Davidson演算法已經在數值上，證明是一種具有發展潛力的數值方法。多項式Jacobi-Davidson演算法是一種子空間方法，該方法從搜尋空間截取出近似特徵向量與特徵值以及透過對修正方程求解以延展搜尋空間的維度。本計畫第一個目標是開發，實現，並研究平行化多層多項式JD演算法，重點的應用是流體結構交互作用特徵值問題。利用多層次的慨念建構出較好搜尋空間的基底。另一方面，專為校正方程設計的多層次Schwarz preconditioner，預期能提高多項式JD演算法的可擴展性，可擴展性是平行演算法對於大規模計算並利用大量的處理器所必須具備的重要的性質。在這個計畫的第二個目的是提供一個用戶友善的科學軟體包括對科學家和工程師，以滿足其對大規模多項式特徵問題求解的需求而能利用最少編程開發的時間完成複雜系統的數值模擬。為了實現這一目標，我們計劃進行廣泛的數值實驗，以確定算法中最佳參數的設定為預設值，並希望能以數值結果以證明顯示其與其他最先進的特徵值的軟件比較的優越性。此外，一些新的功能，計劃將增加以適用於各種應用。 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. 研究期間：10008 ~ 10107
