Uniformly convergent high order finite element solutions of a singularly perturbed reaction-diffusion equation using mesh equidistribution

G. Beckett, J.A. Mackenzie

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

We study the numerical approximation of a singularly perturbed reaction-diffusion equation using a pth order Galerkin finite element method on a non-uniform grid. The grid is constructed by equidistributing a strictly positive monitor function which is a linear combination of a constant floor and a power of the second derivative of a representation of the boundary layers-obtained using a suitable decomposition of the analytical solution. By the appropriate selection of the monitor function parameters we prove that the numerical solution is insensitive to the size of the singular perturbation parameter and achieves the optimal rate of convergence with respect to the mesh density.
LanguageEnglish
Pages31-45
Number of pages14
JournalApplied Numerical Mathematics
Volume39
Issue number1
DOIs
Publication statusPublished - 2001

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High-order Finite Elements
Equidistribution
Finite Element Solution
Singularly Perturbed
Reaction-diffusion Equations
Monitor
Mesh
Non-uniform Grid
Galerkin Finite Element Method
Optimal Rate of Convergence
Strictly positive
Singular Perturbation
Second derivative
Numerical Approximation
Linear Combination
Boundary Layer
Analytical Solution
Boundary layers
Numerical Solution
Grid

Keywords

  • uniform convergence
  • adaptivity
  • equidistribution
  • singular perturbation
  • reaction-diffusion
  • finite element

Cite this

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