Maximum principle for the finite element solution of time-dependent anisotropic diffusion problems

Xianping Li, Weizhang Huang

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

Preservation of the maximum principle is studied for the combination of the linear finite element method in space and the θ -method in time for solving time-dependent anisotropic diffusion problems. It is shown that the numerical solution satisfies a discrete maximum principle when all element angles of the mesh measured in the metric specified by the inverse of the diffusion matrix are nonobtuse, and the time step size is bounded below and above by bounds proportional essentially to the square of the maximal element diameter. The lower bound requirement can be removed when a lumped mass matrix is used. In two dimensions, the mesh and time step conditions can be replaced by weaker Delaunay-type conditions. Numerical results are presented to verify the theoretical findings.

Original languageEnglish (US)
Pages (from-to)1963-1985
Number of pages23
JournalNumerical Methods for Partial Differential Equations
Volume29
Issue number6
DOIs
StatePublished - Nov 2013
Externally publishedYes

Keywords

  • anisotropic diffusion
  • finite element
  • maximum principle
  • time-dependent

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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