Unifying a posteriori error analysis of five piecewise quadratic discretisations for the biharmonic equation

TL;DR

This paper explores how an abstract property (H) influences a posteriori error analysis for various finite element methods solving the biharmonic equation, providing explicit residual-based error estimates.

Contribution

It extends the abstract framework to derive explicit a posteriori error estimates for multiple nonstandard finite element methods for the biharmonic problem.

Findings

Established explicit residual-based a posteriori error estimates.

Applied the framework to Morley, DG, C^0 interior penalty, and over-penalized schemes.

Demonstrated the framework's effectiveness for general source terms in H^{-2}().

Abstract

An abstract property (H) is the key to a complete a priori error analysis in the (discrete) energy norm for several nonstandard finite element methods in the recent work [Lowest-order equivalent nonstandard finite element methods for biharmonic plates, Carstensen and Nataraj, M2AN, 2022]. This paper investigates the impact of (H) to the a posteriori error analysis and establishes known and novel explicit residual-based a posteriori error estimates. The abstract framework applies to Morley, two versions of discontinuous Galerkin, $C^0$ interior penalty, as well as weakly over-penalized symmetric interior penalty schemes for the biharmonic equation with a general source term in $H^{-2}(\Omega)$.