An Augmented Two-Scale Finite Element Method for Eigenvalue Problems

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  • 1 LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
    2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;
    3 School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 102206, China

Received date: 2023-09-25

  Revised date: 2024-01-16

  Accepted date: 2024-01-17

  Online published: 2025-04-21

Supported by

X. Dai, Y. Du, and A. Zhou were partially supported by the National Key R & D Program of China under Grant Nos. 2019YFA0709600 and 2019YFA0709601 and the National Natural Science Foundation of China under Grant No. 12021001. X. Dai was partially supported by the National Natural Science Foundation of China under Grant No. 92270206. F. Liu was partially supported by the National Natural Science Foundation of China (Grant No. 11771467) and the disciplinary funding of Central University of Finance and Economics.

Abstract

In this paper, an augmented two-scale finite element method is proposed for a class of linear and nonlinear eigenvalue problems on tensor-product domains. Through a correction step, the augmented two-scale finite element solution is obtained by solving an eigenvalue problem on a low-dimensional augmented subspace. Theoretical analysis and numerical experiments show that the augmented two-scale finite element solution achieves the same order of accuracy as the standard finite element solution on a fine grid, but the computational cost required by the former solution is much lower than that demanded by the latter. The augmented two-scale finite element method also improves the approximation accuracy of eigenfunctions in the L2(Ω) norm compared with the two-scale finite element method.

Cite this article

Xiaoying Dai, Yunyun Du, Fang Liu, Aihui Zhou . An Augmented Two-Scale Finite Element Method for Eigenvalue Problems[J]. Communications on Applied Mathematics and Computation, 2025 , 7(2) : 663 -688 . DOI: 10.1007/s42967-024-00375-0

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