### 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 language | English |
---|---|

Pages (from-to) | 507-538 |

Number of pages | 32 |

Journal | IMA Journal of Numerical Analysis |

Volume | 23 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jul 2003 |

Externally published | Yes |

### Keywords

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

### Cite this

*IMA Journal of Numerical Analysis*,

*23*(3), 507-538. https://doi.org/10.1093/imanum/23.3.507

}

*IMA Journal of Numerical Analysis*, vol. 23, no. 3, pp. 507-538. https://doi.org/10.1093/imanum/23.3.507

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

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Convergence of a finite-volume mixed finite-element method for an elliptic-hyperbolic system

AU - Droniou, J.

AU - Eymard, R.

AU - Hilhorst, D.

AU - Zhou, X. D.

PY - 2003/7/1

Y1 - 2003/7/1

N2 - 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.

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.

KW - Finite-volume method

KW - Mixed finite-element method

KW - System of a hyperbolic and an elliptic equation

UR - http://www.scopus.com/inward/record.url?scp=0037490648&partnerID=8YFLogxK

U2 - 10.1093/imanum/23.3.507

DO - 10.1093/imanum/23.3.507

M3 - Article

VL - 23

SP - 507

EP - 538

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

SN - 0272-4979

IS - 3

ER -