The maximum principle violations of the mixed-hybrid finite-element method applied to diffusion equations

Hussein Hoteit, R. Mosé*, B. Philippe, Ph Ackerer, J. Erhel

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

Abstract

The abundant literature of finite-element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties. However, some initial-boundary value problems may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth. This intuitive fact of the diffusion process necessitates, when applying numerical methods, varying the mesh size (in time and space) according to the smoothness of the solution. In this work, the numerical behaviour of the time-dependent solutions for such problems during small time duration obtained by using a non-conforming mixed-hybrid finite-element method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin finite element (FE) as well as the finite-difference (FD) methods are checked. Owing to the fact that the mixed methods violate the discrete maximum principle, some numerical experiments showed that the MHFEM leads sometimes to non-physical peaks in the solution. A diffusivity criterion relating the mesh steps for an artificial initial-boundary value problem will be presented. One of the propositions given to avoid any non-physical oscillations is to use the mass-lumping techniques.

Original languageEnglish (US)
Pages (from-to)1373-1390
Number of pages18
JournalInternational Journal for Numerical Methods in Engineering
Volume55
Issue number12
DOIs
StatePublished - Dec 30 2002

Keywords

  • Discrete maximum principle
  • Mass lumping
  • Mixed-hybrid finite-element method
  • Parabolic problem
  • Refinement

ASJC Scopus subject areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics

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