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

Allendes Flores, Alejandro Ignacio and Barrenechea, Gabriel and Hernández, Erwin and Valentin, Frédéric (2011) A two-level enriched finite element method for a mixed problem. Mathematics of Computation, 80. pp. 11-41. ISSN 0025-5718

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    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

    ORCID iDs

    Allendes Flores, Alejandro Ignacio, Barrenechea, Gabriel ORCID logoORCID: https://orcid.org/0000-0003-4490-678X, Hernández, Erwin and Valentin, Frédéric;
    • Item type:Article
      ID code:29059
      Dates:
      Date
      Event
      July 2011
      Published
      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.
      Keywords:Darcy flow, two-level finite element method , mass conservation, Petrov-Galerkin approach, enriched finite element method, Mathematics, Algebra and Number Theory, Computational Mathematics, Applied Mathematics
      Subjects:Science > Mathematics
      Department:Faculty of Science > Mathematics and Statistics
      Depositing user: Pure Administrator
      Date deposited:07 Mar 2011 23:24
      Last modified:09 Sep 2021 01:48
      Related URLs:
      URI:https://strathprints.strath.ac.uk/id/eprint/29059
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