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An hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem

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  • Feng, Jinhua
  • Wang, Shixi
  • Bi, Hai
  • Yang, Yidu

Abstract

The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics, and has wide applications. In this paper, we study an hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem. Based on the work of Gudi et al. [J Sci Comput, 37 (2008)], using piecewise polynomials of degree p≥3, we derive the a priori error estimates of the approximate eigenfunction in the broken H1 norm and L2 norm which are optimal in h and suboptimal in p. When p=2, the approximate eigenfunctions converge but with only suboptimal convergence rate. When p≥2 and the eigenfunction u∈Hs(Ω)(s≥p+1), we prove that the convergence rate of approximate eigenvalues reaches 2p−2 in h and p−(2s−7) in p. We also discuss the a posterior error estimates of the approximate eigenvalues and implement the adaptive calculation. Numerical experiments show that the methods are easy to implement and can efficiently compute biharmonic eigenvalues.

Suggested Citation

  • Feng, Jinhua & Wang, Shixi & Bi, Hai & Yang, Yidu, 2023. "An hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    • Handle: RePEc:eee:apmaco:v:450:y:2023:i:c:s0096300323001388
    DOI: 10.1016/j.amc.2023.127969
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    References listed on IDEAS

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    1. Yang, Yidu & Han, Jiayu & Bi, Hai & Li, Hao & Zhang, Yu, 2020. "Mixed methods for the elastic transmission eigenvalue problem," Applied Mathematics and Computation, Elsevier, vol. 374(C).
    2. Meng, Jian & Mei, Liquan, 2020. "Discontinuous Galerkin methods of the non-selfadjoint Steklov eigenvalue problem in inverse scattering," Applied Mathematics and Computation, Elsevier, vol. 381(C).
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