A posteriori error estimates for nonconforming discretizations of singularly perturbed biharmonic operators

TL;DR

This paper introduces a residual-based a posteriori error estimator for nonconforming finite element methods applied to singularly perturbed biharmonic problems, ensuring reliability and efficiency up to polynomial approximation errors.

Contribution

It develops a new error estimator applicable to various nonconforming elements for biharmonic equations, accounting for polynomial degree variations and right-hand side oscillations.

Findings

Estimator is reliable and locally efficient.

Applicable to many existing nonconforming finite elements.

Accounts for polynomial degree and oscillation effects.

Abstract

For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local best-approximation error of the finite element function by piecewise polynomial functions of the degree determining the expected approximation order, which need not coincide with the maximal polynomial degree of the element, for example if bubble functions are used. The error estimator is shown to be reliable and locally efficient up to this polynomial best-approximation error and oscillations of the right-hand side.