Local and Parallel Multigrid Method for Nonlinear Eigenvalue Problems

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In this paper, a new type of local and parallel algorithm is proposed to solve nonlinear eigenvalue problem based on multigrid discretization. Instead of solving the nonlinear eigenvalue problem directly in each level mesh, our method converts the nonlinear eigenvalue problem in the finest mesh to a linear boundary value problem on each level mesh and some nonlinear eigenvalue problems on the coarsest mesh. Further, the involved linear boundary value problems are solved using the local and parallel strategy. As no nonlinear eigenvalue problem is being solved directly on the fine spaces, which is time-consuming, this new type of local and parallel multigrid method evidently improves the efficiency of nonlinear eigenvalue problem solving. We provide a rigorous theoretical analysis for our algorithm and present details on numerical simulations to support our theory.

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Correspondence to Fei Xu.

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This work was supported by National Natural Science Foundation of China (Grant Nos. 11801021, 11971047), Foundation for Fundamental Research of Beijing University of Technology (No. 006000546318504)

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Xu, F., Huang, Q. Local and Parallel Multigrid Method for Nonlinear Eigenvalue Problems. J Sci Comput 82, 20 (2020) doi:10.1007/s10915-020-01128-w

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  • Nonlinear eigenvalue problem
  • Local and parallel
  • Finite element method
  • Multilevel correction method