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

Alejandro Ignacio Allendes Flores, Gabriel Barrenechea, Erwin Hernández , Frédéric Valentin

Research output: Contribution to journalArticle

3 Citations (Scopus)

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
LanguageEnglish
Pages11-41
Number of pages31
JournalMathematics of Computation
Volume80
DOIs
Publication statusPublished - Jul 2011

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Mixed Problem
Finite Element Method
Darcy Equation
Two-level Method
Finite element method
Interpolation Spaces
Petrov-Galerkin
Optimal Error Estimates
Numerical Approximation
Well-posedness
Basis Functions
Numerical analysis
Numerical Analysis
Optimality
Interpolation
Uniqueness
Degree of freedom
Numerical Experiment
Finite Element
Norm

Keywords

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

Cite this

Allendes Flores, Alejandro Ignacio ; Barrenechea, Gabriel ; Hernández , Erwin ; Valentin, Frédéric. / A two-level enriched finite element method for a mixed problem. In: Mathematics of Computation. 2011 ; Vol. 80. pp. 11-41.
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A two-level enriched finite element method for a mixed problem. / Allendes Flores, Alejandro Ignacio; Barrenechea, Gabriel; Hernández , Erwin; Valentin, Frédéric.

In: Mathematics of Computation, Vol. 80, 07.2011, p. 11-41.

Research output: Contribution to journalArticle

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