A GENERALIZED MIMETIC FINITE DIFFERENCE METHOD AND TWO-POINT FLUX SCHEMES OVER VORONOI DIAGRAMS

Omar Al-Hinai; Mary F. Wheeler; Ivan Yotov

doi:10.1051/m2an/2016033

A GENERALIZED MIMETIC FINITE DIFFERENCE METHOD AND TWO-POINT FLUX SCHEMES OVER VORONOI DIAGRAMS

Omar Al-Hinai, Mary F. Wheeler, Ivan Yotov

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

12 Scopus citations

Abstract

We develop a generalization of the mimetic finite difference (MFD) method for second order elliptic problems that extends the family of convergent schemes to include two-point flux approximation (TPFA) methods over general Voronoi meshes, which are known to satisfy the discrete maximum principle. The method satisfies a modified consistency condition, which utilizes element and face weighting functions. This results in shifting the points on the elements and faces where the pressure and the flux are most accurately approximated. The flux bilinear form is non-symmetric in general, although it reduces to a symmetric form in the case of TPFA. It can be defined as the L2-inner product of vectors in two H(Ω;div) discrete spaces, which are constructed via suitable lifting operators. A specific construction of such lifting operators is presented on rectangles. We note that a different choice is made for test and trial spaces, therefore the method can be viewed as a H(Ω;div)-conforming Petrov-Galerkin Mixed Finite Element method. We prove first-order convergence in pressure and flux, and superconvergence of the pressure under further restrictions. We present numerical results that support the theory.

Original language	English (US)
Pages (from-to)	679-706
Number of pages	28
Journal	ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
Volume	51
Issue number	2
DOIs	https://doi.org/10.1051/m2an/2016033
State	Published - Feb 24 2017
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2022-06-03
Acknowledgements: The authors acknowledge the financial support from the King Abdullah University of Science and Technology Academic Excellence Alliance, from Saudi Aramco and from the Center for Subsurface Modeling Industrial Affiliates. The second author was partially supported by DOE Grant DE-FG02-04ER25617; the third author was partially supported by DOE Grant DE-FG02-04ER25618 and NSF Grant DMS 1418947. We also acknowledge helpful feedback from Todd Arbogast, Mojdeh Delshad, Leszek Demkowicz, Xin Yang, Guangri (Gary) Xue and Jerome Droniou.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Modeling and Simulation
Analysis
Applied Mathematics
General Mathematics
Numerical Analysis

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10.1051/m2an/2016033

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title = "A GENERALIZED MIMETIC FINITE DIFFERENCE METHOD AND TWO-POINT FLUX SCHEMES OVER VORONOI DIAGRAMS",

abstract = "We develop a generalization of the mimetic finite difference (MFD) method for second order elliptic problems that extends the family of convergent schemes to include two-point flux approximation (TPFA) methods over general Voronoi meshes, which are known to satisfy the discrete maximum principle. The method satisfies a modified consistency condition, which utilizes element and face weighting functions. This results in shifting the points on the elements and faces where the pressure and the flux are most accurately approximated. The flux bilinear form is non-symmetric in general, although it reduces to a symmetric form in the case of TPFA. It can be defined as the L2-inner product of vectors in two H(Ω;div) discrete spaces, which are constructed via suitable lifting operators. A specific construction of such lifting operators is presented on rectangles. We note that a different choice is made for test and trial spaces, therefore the method can be viewed as a H(Ω;div)-conforming Petrov-Galerkin Mixed Finite Element method. We prove first-order convergence in pressure and flux, and superconvergence of the pressure under further restrictions. We present numerical results that support the theory.",

author = "Omar Al-Hinai and Wheeler, {Mary F.} and Ivan Yotov",

note = "KAUST Repository Item: Exported on 2022-06-03 Acknowledgements: The authors acknowledge the financial support from the King Abdullah University of Science and Technology Academic Excellence Alliance, from Saudi Aramco and from the Center for Subsurface Modeling Industrial Affiliates. The second author was partially supported by DOE Grant DE-FG02-04ER25617; the third author was partially supported by DOE Grant DE-FG02-04ER25618 and NSF Grant DMS 1418947. We also acknowledge helpful feedback from Todd Arbogast, Mojdeh Delshad, Leszek Demkowicz, Xin Yang, Guangri (Gary) Xue and Jerome Droniou. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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N1 - KAUST Repository Item: Exported on 2022-06-03 Acknowledgements: The authors acknowledge the financial support from the King Abdullah University of Science and Technology Academic Excellence Alliance, from Saudi Aramco and from the Center for Subsurface Modeling Industrial Affiliates. The second author was partially supported by DOE Grant DE-FG02-04ER25617; the third author was partially supported by DOE Grant DE-FG02-04ER25618 and NSF Grant DMS 1418947. We also acknowledge helpful feedback from Todd Arbogast, Mojdeh Delshad, Leszek Demkowicz, Xin Yang, Guangri (Gary) Xue and Jerome Droniou. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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N2 - We develop a generalization of the mimetic finite difference (MFD) method for second order elliptic problems that extends the family of convergent schemes to include two-point flux approximation (TPFA) methods over general Voronoi meshes, which are known to satisfy the discrete maximum principle. The method satisfies a modified consistency condition, which utilizes element and face weighting functions. This results in shifting the points on the elements and faces where the pressure and the flux are most accurately approximated. The flux bilinear form is non-symmetric in general, although it reduces to a symmetric form in the case of TPFA. It can be defined as the L2-inner product of vectors in two H(Ω;div) discrete spaces, which are constructed via suitable lifting operators. A specific construction of such lifting operators is presented on rectangles. We note that a different choice is made for test and trial spaces, therefore the method can be viewed as a H(Ω;div)-conforming Petrov-Galerkin Mixed Finite Element method. We prove first-order convergence in pressure and flux, and superconvergence of the pressure under further restrictions. We present numerical results that support the theory.

AB - We develop a generalization of the mimetic finite difference (MFD) method for second order elliptic problems that extends the family of convergent schemes to include two-point flux approximation (TPFA) methods over general Voronoi meshes, which are known to satisfy the discrete maximum principle. The method satisfies a modified consistency condition, which utilizes element and face weighting functions. This results in shifting the points on the elements and faces where the pressure and the flux are most accurately approximated. The flux bilinear form is non-symmetric in general, although it reduces to a symmetric form in the case of TPFA. It can be defined as the L2-inner product of vectors in two H(Ω;div) discrete spaces, which are constructed via suitable lifting operators. A specific construction of such lifting operators is presented on rectangles. We note that a different choice is made for test and trial spaces, therefore the method can be viewed as a H(Ω;div)-conforming Petrov-Galerkin Mixed Finite Element method. We prove first-order convergence in pressure and flux, and superconvergence of the pressure under further restrictions. We present numerical results that support the theory.

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A GENERALIZED MIMETIC FINITE DIFFERENCE METHOD AND TWO-POINT FLUX SCHEMES OVER VORONOI DIAGRAMS

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