A study on nonnegativity preservation in finite element approximation of Nagumo-type nonlinear differential equations

Xianping Li, Weizhang Huang

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Preservation of nonnengativity and boundedness in the finite element solution of Nagumo-type equations with general anisotropic diffusion is studied. Linear finite elements and the backward Euler scheme are used for the spatial and temporal discretization, respectively. An explicit, an implicit, and two hybrid explicit–implicit treatments for the nonlinear reaction term are considered. Conditions for the mesh and the time step size are developed for the numerical solution to preserve nonnegativity and boundedness. The effects of lumping of the mass matrix and the reaction term are also discussed. The analysis shows that the nonlinear reaction term has significant effects on the conditions for both the mesh and the time step size. Numerical examples are given to demonstrate the theoretical findings.

Original languageEnglish (US)
Pages (from-to)49-67
Number of pages19
JournalApplied Mathematics and Computation
Volume309
DOIs
StatePublished - Sep 15 2017
Externally publishedYes

Keywords

  • Anisotropic diffusion
  • Boundedness preservation
  • Finite element method
  • Maximum principle
  • Nagumo equation
  • Nonnegativity preservation

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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