Adaptive Morley element algorithms for the biharmonic eigenvalue problem
Springer Science and Business Media LLC -- Journal of Inequalities and Applications
DOI 10.1186/s13660-018-1643-9
Keyword(s)
  1. 65N25
  2. 65N30
  3. 65N15
  4. Biharmonic eigenvalues
  5. Morley elements
  6. Adaptive algorithms
  7. An inequality on Rayleigh quotient
Abstract(s)

This paper is devoted to the adaptive Morley element algorithms for a biharmonic eigenvalue problem in [TeX:] \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}$\end{document}Rn ([TeX:] \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq2$\end{document}n2). We combine the Morley element method with the shifted-inverse iteration including Rayleigh quotient iteration and the inverse iteration with fixed shift to propose multigrid discretization schemes in an adaptive fashion. We establish an inequality on Rayleigh quotient and use it to prove the efficiency of the adaptive algorithms. Numerical experiments show that these algorithms are efficient and can get the optimal convergence rate.