Morley finite element method for the eigenvalues of the biharmonic operator

Abstract

This paper studies the nonconforming Morley finite element approximation of the eigenvalues of the biharmonic operator. A new C 1 conforming companion operator leads to an L 2 error estimate for the Morley finite element method which directly compares the L 2 error with the error in the energy norm and, hence, can dispense with any additional regularity assumptions. Furthermore, the paper presents new eigenvalue error estimates for nonconforming finite elements that bound the error of (possibly multiple or clustered) eigenvalues by the approximation error of the computed invariant subspace. An application is the proof of optimal convergence rates for the adaptive Morley finite element method for eigenvalue clusters.

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

“…There has been a long history on developing the finite element methods of the biharmonic eigenvalue problem, and many schemes have been proposed for discretization [9,11,25,36], computation of guaranteed upper and lower bounds [10,22,23,43], and adaptive method and its convergence analysis [17]. This paper is devoted to studying the multilevel efficient method of the biharmonic eigenvalue problem.…”

Saturation and Reliable Hierarchical a Posteriori Morley Finite Element Error Control

“…The proof of Theorem 2.1 is based on the following two lemmas. The first lemma states quasi-orthogonality which has been proven by [20] and [13].…”

Stable splitting of polyharmonic operators by generalized Stokes systems

“…The a priori error estimate can be deduced with the techniques of [12] for d = 2 and [16] for d = 3.…”