Sandpiles and superconductors: Nonconforming linear finite element approximations for mixed formulations of quasi-variational inequalities

John W. Barrett, Leonid Prigozhin

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

Abstract

Similar evolutionary variational and quasi-variational inequalities with gradient constraints arise in the modelling of growing sandpiles and type-II superconductors. Recently, mixed formulations of these inequalities were used for establishing existence results in the quasi-variational inequality case. Such formulations, and this is an additional advantage, made it possible to determine numerically not only the primal variables, e.g., the evolving sand surface and the magnetic field for sandpiles and superconductors, respectively, but also the dual variables, the sand flux and the electric field. Numerical approximations of these mixed formulations in previous studies employed the Raviart-Thomas element of the lowest order. Here we introduce simpler numerical approximations of these mixed formulations based on the nonconforming linear finite element. We prove (subsequence) convergence of these approximations and illustrate their effectiveness by numerical experiments.

Original languageEnglish
Pages (from-to)1-38
Number of pages38
JournalIMA Journal of Numerical Analysis
Volume35
Issue number1
DOIs
StatePublished - 1 Jan 2015

Keywords

  • convergence analysis
  • critical state problems
  • nonconforming finite elements
  • power laws
  • primal and mixed formulations
  • quasi-variational inequalities

ASJC Scopus subject areas

  • General Mathematics
  • Computational Mathematics
  • Applied Mathematics

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