High-order mass-lumped schemes for nonlinear degenerate elliptic equations

Jerome Droniou, Robert Eymard

Research output: Contribution to journalArticleResearchpeer-review

Abstract

We present and analyze a numerical framework for the approximation of nonlinear degenerate elliptic equations of the Stefan or porous medium types. This framework is based on piecewise constant approximations for the functions, which we show are essentially necessary to obtain convergence and error estimates. Convergence is established without regularity assumption on the solution. A detailed analysis is then performed to understand the design properties that enable a scheme, despite these piecewise constant approximations and the degeneracy of the model, to satisfy high-order error estimates if the solution is piecewise smooth. Numerical tests, based on continuous and discontinuous approximation methods, are provided on a variety of one- and twodimensional problems, showing the influence on the convergence rate of the nature of the degeneracy and of the design choices.

Original languageEnglish
Pages (from-to)153-188
Number of pages36
JournalSIAM Journal on Numerical Analysis
Volume58
Issue number1
DOIs
Publication statusPublished - 8 Jan 2020

Keywords

  • Discontinuous Galerkin
  • Error estimate
  • Finite elements
  • Gradient discretization method
  • Mass-lumping
  • Nonlinear degenerate elliptic equations
  • Numerical scheme
  • Porous medium equation
  • Stefan problem

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