### Abstract

Original language | English |
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Pages (from-to) | 31-45 |

Number of pages | 14 |

Journal | Applied Numerical Mathematics |

Volume | 39 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2001 |

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### Keywords

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

### Cite this

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**Uniformly convergent high order finite element solutions of a singularly perturbed reaction-diffusion equation using mesh equidistribution.** / Beckett, G.; Mackenzie, J.A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Uniformly convergent high order finite element solutions of a singularly perturbed reaction-diffusion equation using mesh equidistribution

AU - Beckett, G.

AU - Mackenzie, J.A.

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

KW - uniform convergence

KW - adaptivity

KW - equidistribution

KW - singular perturbation

KW - reaction-diffusion

KW - finite element

UR - http://dx.doi.org/10.1016/S0168-9274(01)00049-6

U2 - 10.1016/S0168-9274(01)00049-6

DO - 10.1016/S0168-9274(01)00049-6

M3 - Article

VL - 39

SP - 31

EP - 45

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

IS - 1

ER -