Nonconforming Finite Element Discretisation for Semilinear Problems with Trilinear Nonlinearity

TL;DR

This paper develops a comprehensive error analysis framework for nonconforming finite element methods, especially Morley FEM, applied to semilinear problems with trilinear nonlinearities, without small data assumptions.

Contribution

It introduces new stability and error estimates for Morley and Crouzeix-Raviart FEMs in semilinear elliptic problems, extending analysis to nonconforming discretizations.

Findings

Established best-approximation a priori error bounds.

Derived residual-based a posteriori error estimates.

Identified parameters ensuring error control for various discretizations.

Abstract

The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of 2D Navier-Stokes problem and in the von K\'{a}rm\'{a}n equations. This paper establishes a best-approximation a~priori error analysis and an a~posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity. The analysis avoids any smallness assumptions on the data and so has to provide discrete stability by a perturbation analysis before the Newton-Kantorovic theorem can provide the existence of discrete solutions. An abstract framework for the stability analysis in terms of discrete operators from the medius analysis leads to new results on the nonconforming Crouzeix-Raviart FEM for second-order linear non-selfadjoint and indefinite elliptic problems with $L^\infty$ coefficients. The paper identifies six parameters and sufficient conditions for the local a~priori and a~posteriori error control of conforming and nonconforming discretisations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications. This leads to new best-approximation error estimates and to a~posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM.