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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.

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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.