Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations

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Abstract

We give a counterexample of an anisotropic triangulation on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. Our example is given in the context of a singularly perturbed reaction-diffusion equation, whose exact solution exhibits a sharp boundary layer. Furthermore, we give a theoretical justification of the observed numerical phenomena using a finite-difference representation of the considered finite element methods. Both standard and lumped-mass cases are addressed.

Original languageEnglish
Pages (from-to)2061-2070
Number of pages10
JournalMathematics of Computation
Volume83
Issue number289
DOIs
Publication statusPublished - 2014

Keywords

  • Anisotropic triangulation
  • Bakhvalov mesh
  • Linear finite elements
  • Maximum norm
  • Shishkin mesh
  • Singular perturbation

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