On nonnegativity preservation in finite element methods for subdiffusion equations

Bangti Jin, Raytcho Lazarov, Vidar Thomée, Zhi Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

10 Citations (Scopus)

Abstract

We consider three types of subdiffusion models, namely singleterm, multi-term and distributed order fractional diffusion equations, for which the maximum-principle holds and which, in particular, preserve nonnegativity. Hence the solution is nonnegative for nonnegative initial data. Following earlier work on the heat equation, our purpose is to study whether this property is inherited by certain spatially semidiscrete and fully discrete piecewise linear finite element methods, including the standard Galerkin method, the lumped mass method and the finite volume element method. It is shown that, as for the heat equation, when the mass matrix is nondiagonal, nonnegativity is not preserved for small time or time-step, but may reappear after a positivity threshold. For the lumped mass method nonnegativity is preserved if and only if the triangulation in the finite element space is of Delaunay type. Numerical experiments illustrate and complement the theoretical results.
Original languageEnglish
Pages (from-to)2239-2260
Number of pages22
JournalMathematics of Computation
Volume86
Issue number307
DOIs
Publication statusPublished - 1 Jan 2017
Externally publishedYes

Keywords

  • Caputo fractional derivative
  • Finite element method
  • Nonnegativity preservation
  • Subdiffusion

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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