The Hessian Discretisation Method for Fourth Order Linear Elliptic Equations

Jérôme Droniou, Bishnu P. Lamichhane, Devika Shylaja

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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 P1 finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme and for a finite volume scheme.

Original languageEnglish
Pages (from-to)1405-1437
Number of pages33
JournalJournal of Scientific Computing
Volume78
Issue number3
DOIs
Publication statusPublished - 2019

Keywords

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

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