| Summary: | 碩士 === 國立中央大學 === 數學研究所 === 100 === Many scientific and engineering applications require accurate, fast, robust, and scalable numerical solution of large sparse algebraic polynomial eigenvalue problems (PEVPs) arising from some appropriate discretization 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 for finding their interior spectrum of the PEVPs. 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. In this research, we propose the two-level PJD algorithm for PEVPs with emphasis on the application of the dissipative acoustic cubic eigenvalue problem. The proposed two-level 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 low-cost and efficient preconditioner based on Schwarz framework, coarse restricted additive Schwarz (RAS_c) preconditioner for the correction equation, which plays a crucial role in parallel computing for large-scale problems by using a large number of processors. Some numerical examples obtained on a parallel cluster of computers are given to demonstrate the robustness and scalability of our PJD algorithm.
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