Improved L2 estimate for gradient schemes and super-convergence of the TPFA finite volume scheme

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6 Citations (Scopus)

Abstract

The gradient discretization method is a generic framework that is applicable to a number of schemes for diffusion equations, and provides in particular generic error estimates in L2 and H1-like norms. In this article, we establish an improved L2 error estimate for gradient schemes. This estimate is applied to a family of gradient schemes, namely the hybrid mimetic mixed (HMM) schemes, and yields an O(h2) superconvergence rate in L2 norm, provided local compensations occur between the cell points used to define the scheme and the centers of mass of the cells. To establish this result, a modified HMM method is designed by just changing the quadrature of the source term; this modified HMM enjoys a super-convergence result even on meshes without local compensations. Finally, the link between HMM and two-point flux approximation (TPFA) finite volume schemes is exploited to partially answer a long-standing conjecture on the super-convergence of TPFA schemes.

Original languageEnglish
Pages (from-to)1254-1293
Number of pages40
JournalIMA Journal of Numerical Analysis
Volume38
Issue number3
DOIs
Publication statusPublished - 17 Jul 2018

Keywords

  • super-convergence
  • two-point flux approximation finite volumes
  • hybrid mimetic mixed methods
  • gradient schemes

Cite this

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title = "Improved L2 estimate for gradient schemes and super-convergence of the TPFA finite volume scheme",
abstract = "The gradient discretization method is a generic framework that is applicable to a number of schemes for diffusion equations, and provides in particular generic error estimates in L2 and H1-like norms. In this article, we establish an improved L2 error estimate for gradient schemes. This estimate is applied to a family of gradient schemes, namely the hybrid mimetic mixed (HMM) schemes, and yields an O(h2) superconvergence rate in L2 norm, provided local compensations occur between the cell points used to define the scheme and the centers of mass of the cells. To establish this result, a modified HMM method is designed by just changing the quadrature of the source term; this modified HMM enjoys a super-convergence result even on meshes without local compensations. Finally, the link between HMM and two-point flux approximation (TPFA) finite volume schemes is exploited to partially answer a long-standing conjecture on the super-convergence of TPFA schemes.",
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Improved L2 estimate for gradient schemes and super-convergence of the TPFA finite volume scheme. / Droniou, Jerome Daniel Raymond; Nataraj, Neela.

In: IMA Journal of Numerical Analysis, Vol. 38, No. 3, 17.07.2018, p. 1254-1293.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - Improved L2 estimate for gradient schemes and super-convergence of the TPFA finite volume scheme

AU - Droniou, Jerome Daniel Raymond

AU - Nataraj, Neela

PY - 2018/7/17

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N2 - The gradient discretization method is a generic framework that is applicable to a number of schemes for diffusion equations, and provides in particular generic error estimates in L2 and H1-like norms. In this article, we establish an improved L2 error estimate for gradient schemes. This estimate is applied to a family of gradient schemes, namely the hybrid mimetic mixed (HMM) schemes, and yields an O(h2) superconvergence rate in L2 norm, provided local compensations occur between the cell points used to define the scheme and the centers of mass of the cells. To establish this result, a modified HMM method is designed by just changing the quadrature of the source term; this modified HMM enjoys a super-convergence result even on meshes without local compensations. Finally, the link between HMM and two-point flux approximation (TPFA) finite volume schemes is exploited to partially answer a long-standing conjecture on the super-convergence of TPFA schemes.

AB - The gradient discretization method is a generic framework that is applicable to a number of schemes for diffusion equations, and provides in particular generic error estimates in L2 and H1-like norms. In this article, we establish an improved L2 error estimate for gradient schemes. This estimate is applied to a family of gradient schemes, namely the hybrid mimetic mixed (HMM) schemes, and yields an O(h2) superconvergence rate in L2 norm, provided local compensations occur between the cell points used to define the scheme and the centers of mass of the cells. To establish this result, a modified HMM method is designed by just changing the quadrature of the source term; this modified HMM enjoys a super-convergence result even on meshes without local compensations. Finally, the link between HMM and two-point flux approximation (TPFA) finite volume schemes is exploited to partially answer a long-standing conjecture on the super-convergence of TPFA schemes.

KW - super-convergence

KW - two-point flux approximation finite volumes

KW - hybrid mimetic mixed methods

KW - gradient schemes

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