The Hessian discretisation method for fourth order linear elliptic equations

Jerome Droniou, Bishnu P. Lamichhane, Devika Shylaja

Published 2018-03-19Version 1

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 method (conforming and non-conforming) as well as finite volume method. Finally, we 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.