The Hessian discretisation method for fourth order linear elliptic equations
For researchers in numerical analysis, this work provides a unifying theoretical framework that simplifies the analysis of diverse numerical methods for fourth-order problems, though it is primarily a methodological contribution rather than a breakthrough in performance.
The paper introduces the Hessian discretisation method (HDM), a unified framework for analyzing and designing numerical methods for fourth-order linear elliptic equations. It provides error estimates based on intrinsic accuracy indicators and shows that HDM encompasses many existing methods, while also enabling the design of a novel conforming P1 finite element scheme with gradient recovery.
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.