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Journal of Scientific Computing

, Volume 78, Issue 3, pp 1405–1437 | Cite as

The Hessian Discretisation Method for Fourth Order Linear Elliptic Equations

  • Jérôme Droniou
  • Bishnu P. Lamichhane
  • Devika ShylajaEmail author
Article
  • 70 Downloads

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.

Keywords

Fourth order elliptic equations Numerical schemes Error estimates Hessian discretisation method Hessian schemes Finite element method Finite volume method Gradient recovery method 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityClaytonAustralia
  2. 2.School of Mathematical and Physical SciencesUniversity of NewcastleCallaghanAustralia
  3. 3.IITB-Monash Research AcademyIndian Institute of Technology BombayPowaiIndia

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