The Hessian discretisation method for fourth order linear elliptic equations

TL;DR

This paper introduces the Hessian discretisation method (HDM), a unified framework for analyzing and designing numerical schemes for fourth order linear elliptic equations, encompassing various existing methods and enabling new scheme development.

Contribution

The paper presents the HDM framework that unifies and generalizes many existing numerical methods for fourth order elliptic problems, and introduces a new conforming finite element-based scheme.

Findings

Error estimates based on intrinsic indicators

HDM covers finite element and finite volume methods

Numerical experiments validate the new scheme and existing methods

Abstract

In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the considered model. An error estimate is obtained, using only these intrinsic indicators, when the HDM framework is applied to linear fourth order problems. It is shown that HDM encompasses a large number of numerical methods for fourth order elliptic problems: finite element methods (conforming and non-conforming) as well as finite volume methods. We also use the HDM to design a novel method, based on conforming $\mathbb{P}_1$ finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme and for a finite volume scheme.