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A priori and a posteriori error analysis for discontinuous Galerkin finite element approximations of biharmonic eigenvalue problems

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Abstract

In this paper, we express and analyze mixed discontinuous Galerkin(DG) methods of biharmonic eigenvalue problems as well as present the error analysis for them. The analysis consists of two parts. First, we propose a residual-based a posteriori error estimator in the approximate eigenfunctions and eigenvalues. The error in the eigenfunctions is measured both in the L2 and DG (energy-like) norms. In addition, we prove that if the error estimator converges to zero, then the distance of the computed eigenfunction from the true eigenspace also converges to zero, and so, the computed eigenvalue converges to a true eigenvalue. Next, we establish an a priori error estimate with the optimal convergence order both in the L2 and DG norms. We show that the methods can retain the same convergence properties they enjoy in the case of source problems.

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Authors and Affiliations

  1. Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

    Liang Wang, Chunguang Xiong & Huibin Wu

  2. Third Institute of Oceanography, State Oceanic Administration, China No. 178 Daxue Road, Siming District, Xiamen, 361005, China

    Fusheng Luo

Authors
  1. Liang Wang
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  2. Chunguang Xiong
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  3. Huibin Wu
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  4. Fusheng Luo
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Correspondence to Chunguang Xiong.

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Communicated by: Ilaria Perugia

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subject is supported partially by NSFC No. 10871218.

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Wang, L., Xiong, C., Wu, H. et al. A priori and a posteriori error analysis for discontinuous Galerkin finite element approximations of biharmonic eigenvalue problems. Adv Comput Math 45, 2623–2646 (2019). https://doi.org/10.1007/s10444-019-09689-7

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  • DOI: https://doi.org/10.1007/s10444-019-09689-7

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