Morley Finite Element Method for the Eigenvalues of the Biharmonic Operator
Provides rigorous error analysis for nonconforming finite elements in eigenvalue problems, benefiting numerical analysts working on adaptive methods for biharmonic operators.
This paper develops new error estimates for the Morley finite element method applied to biharmonic eigenvalue problems, achieving optimal convergence rates for eigenvalue clusters without additional regularity assumptions.
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.