A symmetric nodal conservative finite element method for the Darcy equation

G. R. Barrenechea, L. P. Franca, F. Valentin

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Abstract

This work introduces and analyzes novel stable Petrov-Galerkin EnrichedMethods (PGEM) for the Darcy problem based on the simplest but unstable continuous P1/P0 pair. Stability is recovered inside a Petrov-Galerkin framework where element-wise dependent residual functions, named multi-scale functions, enrich both velocity and pressure trial spaces. Unlike the velocity test space that is augmented with bubble-like functions, multi-scale functions correct edge residuals as well. The multi-scale functions turn out to be the well-known lowest order Raviart-Thomas basis functions for the velocity and discontinuous quadratics polynomial functions for the pressure. The enrichment strategy suggests the way to recover the local mass conservation property for nodal-based interpolation spaces. We prove that the method and its symmetric version are well-posed and achieve optimal error estimates in natural norms. Numerical validations confirm claimed theoretical results.
Original languageEnglish
Pages (from-to)3652-3677
Number of pages26
JournalSIAM Journal on Numerical Analysis
Volume47
Issue number5
DOIs
Publication statusPublished - 2009

Keywords

  • Darcy model
  • enriched space
  • simplest element
  • Petrov-Galerkin approach

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