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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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A two-level enriched finite element method for a mixed problem
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by Alejandro Allendes, Gabriel R. Barrenechea, Erwin Hernández and Frédéric Valentin PDF
Math. Comp. 80 (2011), 11-41 Request permission


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
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  • Gabriel R. Barrenechea
  • Affiliation: Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom
  • Email:
  • Erwin Hernández
  • Affiliation: Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile
  • Email:
  • Frédéric Valentin
  • Affiliation: Departamento de Matemática Aplicada e Computacional, Laboratório Nacional de Computação Científica, Av. Getúlio Vargas, 333, 25651-070 Petrópolis - RJ, Brazil
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  • 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.
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 80 (2011), 11-41
  • MSC (2010): Primary 65N30, 65N12; Secondary 76S99
  • DOI:
  • MathSciNet review: 2728970