Abstract
This paper is devoted to the adaptive Morley element algorithms for a biharmonic eigenvalue problem in mathbb{R}^{n} (ngeq2). 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.
Highlights
1 Introduction Biharmonic equation/eigenvalue problem plays an important role in elastic mechanics
The Morley element was extended to arbitrarily dimensions by Wang and Xu [2] in 2006
The Morley element has been employed to solve the biharmonic eigenvalue problem, including the vibration of a plate; and [9] studied its a priori error estimate. [10, 11] studied a posteriori error estimate and the adaptive method, [12] adopted a new method dispensing with any additional regularity assumption to study the error estimates and adaptive algorithms
Summary
Biharmonic equation/eigenvalue problem plays an important role in elastic mechanics. In 1968, Morley designed a famous non-conforming element called the Morley element [1] to solve biharmonic equation (plate bending problem). The a priori/a posteriori error estimate was studied in [3,4,5,6] and the convergence and optimality of the adaptive Morley element method was proved in [7, 8]. The Morley element has been employed to solve the biharmonic eigenvalue problem, including the vibration of a plate; and [9] studied its a priori error estimate. [10, 12] have employed the adaptive Morley element algorithms for the biharmonic eigenvalue problem based on solving directly the original eigenvalue problem a(u, v) = λb(u, v) in each iteration. We establish the adaptive Morley element algorithms based on the shifted-inverse iteration including Rayleigh quotient iteration and the inverse iteration with fixed shift to solve the biharmonic eigenvalue problem. Referring to [19], we combine the standard local refined adaptive algorithm with uniformly refined algorithm to give new algorithms
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