Hessian discretisation method for fourth-order semi-linear elliptic equations: applications to the von Kármán and Navier–Stokes models

Abstract

This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth-order semi-linear elliptic equations with trilinear non-linearity. Four properties, namely, the coercivity, consistency, limit-conformity and compactness, enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications, namely, the Navier–Stokes equations in stream function vorticity formulation and the von Kármán equations of plate bending, are discussed. Results of numerical experiments are presented for the Morley and Adini ncFEMs, and the GR method.