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Adaptive Morley element algorithms for the biharmonic eigenvalue problem.

Hao Li1, Yidu Yang1

  • 1The School of the Mathematical Sciences, Guizhou Normal University, Gui Yang, China.

Journal of Inequalities and Applications
|March 16, 2018
PubMed
Summary
This summary is machine-generated.

This study introduces adaptive Morley element algorithms for biharmonic eigenvalue problems. These efficient methods achieve optimal convergence rates for solving complex mathematical equations.

Keywords:
Adaptive algorithmsAn inequality on Rayleigh quotientBiharmonic eigenvaluesMorley elements

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Partial differential equations

Background:

  • Biharmonic eigenvalue problems are crucial in various scientific and engineering fields.
  • Existing numerical methods may lack efficiency or optimal convergence for these problems.
  • Adaptive algorithms offer potential for improved computational performance.

Purpose of the Study:

  • To develop and analyze adaptive Morley element algorithms for a biharmonic eigenvalue problem.
  • To combine the Morley element method with iterative solvers for multigrid discretization.
  • To prove the efficiency and optimal convergence of the proposed adaptive schemes.

Main Methods:

  • Utilizing the Morley element method for discretization.
  • Implementing shifted-inverse iteration, including Rayleigh quotient iteration and fixed-shift inverse iteration.
  • Developing adaptive multigrid discretization schemes.
  • Establishing a key inequality on the Rayleigh quotient to analyze algorithm efficiency.

Main Results:

  • The proposed adaptive Morley element algorithms are shown to be efficient.
  • The algorithms achieve the optimal convergence rate for the biharmonic eigenvalue problem.
  • Numerical experiments validate the theoretical findings.

Conclusions:

  • Adaptive Morley element algorithms provide an efficient approach for biharmonic eigenvalue problems.
  • The combination of Morley elements and iterative methods with adaptivity yields optimal convergence.
  • These findings contribute to the advancement of numerical methods for solving differential equations.