A new type of adaptive multigrid method is presented for multiple eigenvalue problems based on multilevel correction scheme and adaptive multigrid method. Different from the classical adaptive finite element method wh...
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A new type of adaptive multigrid method is presented for multiple eigenvalue problems based on multilevel correction scheme and adaptive multigrid method. Different from the classical adaptive finite element method which requires to solve eigenvalue problems on the adaptively refined triangulations, with our approach we just need to solve several linear boundary value problems in the current refined space and an eigenvalue problem in a very low dimensional space. Further, the involved boundary value problems are solved by an adaptive multigrid iteration. Since there is no eigenvalue problem to be solved on the refined triangulations, which is quite time-consuming, the proposed method can achieve the same efficiency as that of the adaptive multigrid method for the associated linear boundary value problems. Besides, the corresponding convergence and optimal complexity are verified theoretically and demonstrated numerically. (C) 2022 Elsevier B.V. All rights reserved.
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. I...
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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.
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