### Abstract

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

Pages | 11-41 |

Number of pages | 31 |

Journal | Mathematics of Computation |

Volume | 80 |

DOIs | |

Publication status | Published - Jul 2011 |

### Fingerprint

### Keywords

- Darcy flow
- two-level finite element method
- mass conservation
- Petrov-Galerkin approach
- enriched finite element method

### Cite this

*Mathematics of Computation*,

*80*, 11-41. https://doi.org/10.1090/S0025-5718-2010-02364-6

}

*Mathematics of Computation*, vol. 80, pp. 11-41. https://doi.org/10.1090/S0025-5718-2010-02364-6

**A two-level enriched finite element method for a mixed problem.** / Allendes Flores, Alejandro Ignacio; Barrenechea, Gabriel; Hernández , Erwin; Valentin, Frédéric.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A two-level enriched finite element method for a mixed problem

AU - Allendes Flores, Alejandro Ignacio

AU - Barrenechea, Gabriel

AU - Hernández , Erwin

AU - Valentin, Frédéric

N1 - The second author was partially supported by Starter’s Grant, Faculty of Sciences, University of Strathclyde. The third author was supported by CONICYT Chile, through FONDECYT Project No. 1070276 and by Universidad Santa María through project No. DGIP-USM 120851. The fourth author was supported by CNPq /Brazil Grant No. 304051/2006-3, FAPERJ/Brazil Grant No. E-26/100.519/2007.

PY - 2011/7

Y1 - 2011/7

N2 - The simplest pair of spaces is made inf-sup stable for the mixed form of the Darcy equation. The key ingredient is to enhance the finite element spaces inside a Petrov-Galerkin framework with functions satisfying element-wise local Darcy problems with right hand sides depending on the residuals over elements and edges. The enriched method is symmetric, locally mass conservative and keeps the degrees of freedom of the original interpolation spaces. First, we assume local enrichments exactly computed and we prove uniqueness and optimal error estimates in natural norms. Then, a low cost two-level finite element method is proposed to effectively obtain enhancing basis functions. The approach lays on a two-scale numerical analysis and shows that well-posedness and optimality is kept, despite the second level numerical approximation. Several numerical experiments validate the theoretical results and compares (favourably in some cases) our results with the classical Raviart-Thomas element

AB - The simplest pair of spaces is made inf-sup stable for the mixed form of the Darcy equation. The key ingredient is to enhance the finite element spaces inside a Petrov-Galerkin framework with functions satisfying element-wise local Darcy problems with right hand sides depending on the residuals over elements and edges. The enriched method is symmetric, locally mass conservative and keeps the degrees of freedom of the original interpolation spaces. First, we assume local enrichments exactly computed and we prove uniqueness and optimal error estimates in natural norms. Then, a low cost two-level finite element method is proposed to effectively obtain enhancing basis functions. The approach lays on a two-scale numerical analysis and shows that well-posedness and optimality is kept, despite the second level numerical approximation. Several numerical experiments validate the theoretical results and compares (favourably in some cases) our results with the classical Raviart-Thomas element

KW - Darcy flow

KW - two-level finite element method

KW - mass conservation

KW - Petrov-Galerkin approach

KW - enriched finite element method

UR - http://www.ams.org/journals/mcom/2011-80-273/S0025-5718-2010-02364-6/home.html

U2 - 10.1090/S0025-5718-2010-02364-6

DO - 10.1090/S0025-5718-2010-02364-6

M3 - Article

VL - 80

SP - 11

EP - 41

JO - Mathematics of Computation

T2 - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

ER -