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


Authors: Alejandro Allendes, Gabriel R. Barrenechea, Erwin Hernández and Frédéric Valentin
Journal: Math. Comp. 80 (2011), 11-41
MSC (2010): Primary 65N30, 65N12; Secondary 76S99
DOI: https://doi.org/10.1090/S0025-5718-2010-02364-6
Published electronically: July 26, 2010
MathSciNet review: 2728970
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Abstract: The simplest pair of spaces $ \mathbb{P}_1 / \mathbb{P}_0$ 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.


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Additional Information

Alejandro Allendes
Affiliation: Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom
Email: alejandro.allendes-flores@strath.ac.uk

Gabriel R. Barrenechea
Affiliation: Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom
Email: gabriel.barrenechea@strath.ac.uk

Erwin Hernández
Affiliation: Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile
Email: erwin.hernandez@usm.cl

Frédéric Valentin
Affiliation: Departamento de Matemática Aplicada e Computacional, Laboratório Nacional de Computação Científica, Av. Getúlio Vargas, 333, 25651-070 Petrópolis - RJ, Brazil
Email: valentin@lncc.br

DOI: https://doi.org/10.1090/S0025-5718-2010-02364-6
Keywords: Darcy flow, enriched finite element method, Petrov-Galerkin approach, mass conservation, two-level finite element method
Received by editor(s): October 10, 2008
Received by editor(s) in revised form: July 1, 2009
Published electronically: July 26, 2010
Additional Notes: 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.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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