Convergence Analysis of the Hessian Discretisation Method for Fourth Order Semi-linear Elliptic Equations with General Source

TL;DR

This paper provides a comprehensive convergence analysis of the Hessian Discretisation Method (HDM) for fourth-order semilinear elliptic equations, unifying various numerical schemes and deriving explicit error estimates without requiring regularity assumptions.

Contribution

It introduces a unified HDM framework for convergence analysis, deriving error estimates for Adini ncFEM and GR methods for the first time, applicable to multiple schemes without regularity constraints.

Findings

Error estimates for Adini ncFEM and GR methods are derived for the first time.

The HDM framework unifies convergence analysis for conforming and nonconforming schemes.

Numerical experiments validate the performance of the proposed methods.

Abstract

This paper presents a convergence analysis for the Hessian Discretisation Method (HDM) applied to fourth-order semilinear elliptic equations involving a trilinear nonlinearity and general source, based on two complementary approaches. The HDM serves as a unified framework for the convergence analysis of various numerical schemes, including conforming and nonconforming finite element methods (ncFEMs) and gradient recovery (GR) based methods. Error estimates for the Adini ncFEM and GR methods are derived for the first time, which provide an explicit order of convergence. The analysis relies on four key HDM properties along with a suitable companion operator to establish convergence results. Moreover, a convergence analysis is developed within the HDM framework, which does not require additional regularity assumptions on the exact solution or the assumption that the exact solution is regular. The paper further discusses two significant applications: the Navier--Stokes equations in stream function--vorticity formulation and the von K\'{a}rm\'{a}n equations for plate bending. Numerical experiments are provided to demonstrate the performance of the GR method, Morley, and Adini ncFEMs.