A posteriori error control for fourth-order semilinear problems with quadratic nonlinearity

TL;DR

This paper develops a unified a posteriori error analysis for several finite element methods applied to fourth-order semilinear problems, including the 2D Navier-Stokes equations, introducing a smoother that enhances error estimation reliability.

Contribution

It introduces a quasi-optimal smoother that extends the source term and modifies the nonlinear term, enabling reliable error estimates for multiple discretizations of complex 2D fluid and elasticity problems.

Findings

First reliable a posteriori error estimates for 2D Navier-Stokes in stream-function form

Applicability to five finite element methods including Morley and DG

Enhanced error control for fourth-order semilinear problems

Abstract

A general a posteriori error analysis applies to five lowest-order finite element methods for two fourth-order semi-linear problems with trilinear non-linearity and a general source. A quasi-optimal smoother extends the source term to the discrete trial space, and more importantly, modifies the trilinear term in the stream-function vorticity formulation of the incompressible 2D Navier-Stokes and the von K\'{a}rm\'{a}n equations. This enables the first efficient and reliable a posteriori error estimates for the 2D Navier-Stokes equations in the stream-function vorticity formulation for Morley, two discontinuous Galerkin, $C^0$ interior penalty, and WOPSIP discretizations with piecewise quadratic polynomials.