A parallel two-level polynomial Jacobi–Davidson algorithm for large sparse PDE eigenvalue problems

Many scientific and engineering applications require accurate, fast, robust, and scalable numerical solution of large sparse algebraic polynomial eigenvalue problems (PEVP's) that arise from some appropriate discretization of partial differential equations. The polynomial Jacobi-Davidson (PJD) algorithm has been numerically shown as a promising approach for the PEVP's to finding the interior spectrum. The PJD algorithm is a subspace method, which extracts the candidate eigenpair from a search space and the space updated by embedding the solution of the correction equation at the JD iteration. In this research, we develop and study the two-level PJD algorithm for PEVP's with emphasis on the application of the dissipative acoustic cubic eigenvalue problem. The proposed two-level PJD algorithm consists of two important ingredients: A good initial basis for the search space is constructed on the fine-level by using the interpolation of the coarse solution of the same eigenvalue problem in order to enhance the robustness of the algorithm. Also, an efficient and scalable two-level preconditioner based on the Schwarz framework is used for the correction equation. Some numerical examples obtained on a parallel cluster of computers are given in order to demonstrate the robustness and scalability of our PJD algorithm.