A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems

Beirao Da Veiga, Jerome Droniou, Gianmarco Manzini

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32 Citations (Scopus)

Abstract

We study the numerical approximation to the solution of the steady convection-diffusion equation. The diffusion term is discretized by using the hybrid mimetic method (HMM), which is the unified formulation for the hybrid finite-volume (FV) method, the mixed FV method and the mimetic finite-difference method recently proposed in Droniou et al. (2010, Math. Models Methods Appl. Sci., 20, 265-295). In such a setting we discuss several techniques to discretize the convection term that are mainly adapted from the literature on FV or FV schemes. For this family of schemes we provide a full proof of convergence under very general regularity conditions of the solution field and derive an error estimate when the scalar solution is in H-2(Omega). Finally, we compare the performance of these schemes on a set of test cases selected from the literature in order to document the accuracy of the numerical approximation in both diffusion-and convection-dominated regimes. Moreover, we numerically investigate the behaviour of these methods in the approximation of solutions with boundary layers or internal regions with strong gradients.
Original languageEnglish
Pages (from-to)1357 - 1401
Number of pages45
JournalIMA Journal of Numerical Analysis
Volume31
Issue number4
DOIs
Publication statusPublished - 2011
Externally publishedYes

Cite this

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title = "A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems",
abstract = "We study the numerical approximation to the solution of the steady convection-diffusion equation. The diffusion term is discretized by using the hybrid mimetic method (HMM), which is the unified formulation for the hybrid finite-volume (FV) method, the mixed FV method and the mimetic finite-difference method recently proposed in Droniou et al. (2010, Math. Models Methods Appl. Sci., 20, 265-295). In such a setting we discuss several techniques to discretize the convection term that are mainly adapted from the literature on FV or FV schemes. For this family of schemes we provide a full proof of convergence under very general regularity conditions of the solution field and derive an error estimate when the scalar solution is in H-2(Omega). Finally, we compare the performance of these schemes on a set of test cases selected from the literature in order to document the accuracy of the numerical approximation in both diffusion-and convection-dominated regimes. Moreover, we numerically investigate the behaviour of these methods in the approximation of solutions with boundary layers or internal regions with strong gradients.",
author = "{Da Veiga}, Beirao and Jerome Droniou and Gianmarco Manzini",
year = "2011",
doi = "10.1093/imanum/drq018",
language = "English",
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pages = "1357 -- 1401",
journal = "IMA Journal of Numerical Analysis",
issn = "0272-4979",
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A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems. / Da Veiga, Beirao; Droniou, Jerome; Manzini, Gianmarco.

In: IMA Journal of Numerical Analysis, Vol. 31, No. 4, 2011, p. 1357 - 1401.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems

AU - Da Veiga, Beirao

AU - Droniou, Jerome

AU - Manzini, Gianmarco

PY - 2011

Y1 - 2011

N2 - We study the numerical approximation to the solution of the steady convection-diffusion equation. The diffusion term is discretized by using the hybrid mimetic method (HMM), which is the unified formulation for the hybrid finite-volume (FV) method, the mixed FV method and the mimetic finite-difference method recently proposed in Droniou et al. (2010, Math. Models Methods Appl. Sci., 20, 265-295). In such a setting we discuss several techniques to discretize the convection term that are mainly adapted from the literature on FV or FV schemes. For this family of schemes we provide a full proof of convergence under very general regularity conditions of the solution field and derive an error estimate when the scalar solution is in H-2(Omega). Finally, we compare the performance of these schemes on a set of test cases selected from the literature in order to document the accuracy of the numerical approximation in both diffusion-and convection-dominated regimes. Moreover, we numerically investigate the behaviour of these methods in the approximation of solutions with boundary layers or internal regions with strong gradients.

AB - We study the numerical approximation to the solution of the steady convection-diffusion equation. The diffusion term is discretized by using the hybrid mimetic method (HMM), which is the unified formulation for the hybrid finite-volume (FV) method, the mixed FV method and the mimetic finite-difference method recently proposed in Droniou et al. (2010, Math. Models Methods Appl. Sci., 20, 265-295). In such a setting we discuss several techniques to discretize the convection term that are mainly adapted from the literature on FV or FV schemes. For this family of schemes we provide a full proof of convergence under very general regularity conditions of the solution field and derive an error estimate when the scalar solution is in H-2(Omega). Finally, we compare the performance of these schemes on a set of test cases selected from the literature in order to document the accuracy of the numerical approximation in both diffusion-and convection-dominated regimes. Moreover, we numerically investigate the behaviour of these methods in the approximation of solutions with boundary layers or internal regions with strong gradients.

UR - http://imajna.oxfordjournals.org/content/31/4/1357

U2 - 10.1093/imanum/drq018

DO - 10.1093/imanum/drq018

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VL - 31

SP - 1357

EP - 1401

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

SN - 0272-4979

IS - 4

ER -