### Abstract

the finite element approximation of singularly perturbed reaction–diffusion problems was presented in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) which entailed the solution of an infinite dimensional local boundary value problem. It is not possible to solve this problem exactly and this fact was recognised in the above work where it was indicated that the limitation would be addressed in a subsequent article. We view the present work as fulfilling that promise and as completing the investigation begun in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) by removing the obligation to solve a local problem exactly. The resulting new estimator is indeed fully computable and the first to provide fully computable, robust upper bounds in the setting of singularly perturbed problems discretised by the finite element method

Language | English |
---|---|

Pages | 219-243 |

Number of pages | 25 |

Journal | Numerische Mathematik |

Volume | 119 |

Issue number | 2 |

Early online date | 24 Jun 2011 |

DOIs | |

Publication status | Published - 2011 |

### Fingerprint

### Keywords

- reaction–diffusion problems
- finite elementmethod

### Cite this

*Numerische Mathematik*,

*119*(2), 219-243. https://doi.org/10.1007/s00211-011-0384-1

}

*Numerische Mathematik*, vol. 119, no. 2, pp. 219-243. https://doi.org/10.1007/s00211-011-0384-1

**Fully computable robust a posteriori error bounds for singularly perturbed reaction–diffusion problems.** / Ainsworth, Mark; Vejchodsky, Tomas.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Fully computable robust a posteriori error bounds for singularly perturbed reaction–diffusion problems

AU - Ainsworth, Mark

AU - Vejchodsky, Tomas

PY - 2011

Y1 - 2011

N2 - A procedure for the construction of robust, upper bounds for the error inthe finite element approximation of singularly perturbed reaction–diffusion problems was presented in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) which entailed the solution of an infinite dimensional local boundary value problem. It is not possible to solve this problem exactly and this fact was recognised in the above work where it was indicated that the limitation would be addressed in a subsequent article. We view the present work as fulfilling that promise and as completing the investigation begun in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) by removing the obligation to solve a local problem exactly. The resulting new estimator is indeed fully computable and the first to provide fully computable, robust upper bounds in the setting of singularly perturbed problems discretised by the finite element method

AB - A procedure for the construction of robust, upper bounds for the error inthe finite element approximation of singularly perturbed reaction–diffusion problems was presented in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) which entailed the solution of an infinite dimensional local boundary value problem. It is not possible to solve this problem exactly and this fact was recognised in the above work where it was indicated that the limitation would be addressed in a subsequent article. We view the present work as fulfilling that promise and as completing the investigation begun in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) by removing the obligation to solve a local problem exactly. The resulting new estimator is indeed fully computable and the first to provide fully computable, robust upper bounds in the setting of singularly perturbed problems discretised by the finite element method

KW - reaction–diffusion problems

KW - finite elementmethod

UR - http://www.scopus.com/inward/record.url?scp=80052667051&partnerID=8YFLogxK

U2 - 10.1007/s00211-011-0384-1

DO - 10.1007/s00211-011-0384-1

M3 - Article

VL - 119

SP - 219

EP - 243

JO - Numerische Mathematik

T2 - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 2

ER -