Convergence of a finite-volume mixed finite-element method for an elliptic-hyperbolic system

J. Droniou, R. Eymard, D. Hilhorst, X. D. Zhou

Research output: Contribution to journalArticleResearchpeer-review

4 Citations (Scopus)

Abstract

This paper gives a proof of convergence for the approximate solution of an elliptichyperbolic system, describing the conservation of two immiscible incompressible phases flowing in a porous medium. The approximate solution is obtained by a mixed finite-element method on a large class of meshes for the elliptic equation and a finite-volume method for the hyperbolic equation. Since the considered meshes are not necessarily structured, the proof uses a weak total variation inequality, which cannot yield a BV-estimate. We thus prove, under an L estimate, the weak convergence of the finite-volume approximation. The strong convergence proof is then sketched under regularity assumptions which ensure that the flux is Lipschitz continuous.

Original languageEnglish
Pages (from-to)507-538
Number of pages32
JournalIMA Journal of Numerical Analysis
Volume23
Issue number3
DOIs
Publication statusPublished - 1 Jul 2003
Externally publishedYes

Keywords

  • Finite-volume method
  • Mixed finite-element method
  • System of a hyperbolic and an elliptic equation

Cite this

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Convergence of a finite-volume mixed finite-element method for an elliptic-hyperbolic system. / Droniou, J.; Eymard, R.; Hilhorst, D.; Zhou, X. D.

In: IMA Journal of Numerical Analysis, Vol. 23, No. 3, 01.07.2003, p. 507-538.

Research output: Contribution to journalArticleResearchpeer-review

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AB - This paper gives a proof of convergence for the approximate solution of an elliptichyperbolic system, describing the conservation of two immiscible incompressible phases flowing in a porous medium. The approximate solution is obtained by a mixed finite-element method on a large class of meshes for the elliptic equation and a finite-volume method for the hyperbolic equation. Since the considered meshes are not necessarily structured, the proof uses a weak total variation inequality, which cannot yield a BV-estimate. We thus prove, under an L∞ estimate, the weak convergence of the finite-volume approximation. The strong convergence proof is then sketched under regularity assumptions which ensure that the flux is Lipschitz continuous.

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