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 -inner product of vectors in two 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 -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.
Accepté le :
DOI : 10.1051/m2an/2016033
Mots clés : Mimetic finite difference, finite volume methods, discrete maximum principle, polyhedral meshes, Voronoi diagrams
@article{M2AN_2017__51_2_679_0,
author = {Al-Hinai, Omar and Wheeler, Mary F. and Yotov, Ivan},
title = {A generalized {Mimetic} {Finite} {Difference} method and {Two-Point} {Flux} schemes over {Voronoi} diagrams},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {679--706},
publisher = {EDP-Sciences},
volume = {51},
number = {2},
year = {2017},
doi = {10.1051/m2an/2016033},
mrnumber = {3626415},
zbl = {1398.76148},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2016033/}
}
TY - JOUR AU - Al-Hinai, Omar AU - Wheeler, Mary F. AU - Yotov, Ivan TI - A generalized Mimetic Finite Difference method and Two-Point Flux schemes over Voronoi diagrams JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 679 EP - 706 VL - 51 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016033/ DO - 10.1051/m2an/2016033 LA - en ID - M2AN_2017__51_2_679_0 ER -
%0 Journal Article %A Al-Hinai, Omar %A Wheeler, Mary F. %A Yotov, Ivan %T A generalized Mimetic Finite Difference method and Two-Point Flux schemes over Voronoi diagrams %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 679-706 %V 51 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016033/ %R 10.1051/m2an/2016033 %G en %F M2AN_2017__51_2_679_0
Al-Hinai, Omar; Wheeler, Mary F.; Yotov, Ivan. A generalized Mimetic Finite Difference method and Two-Point Flux schemes over Voronoi diagrams. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 679-706. doi : 10.1051/m2an/2016033. http://www.numdam.org/articles/10.1051/m2an/2016033/
, An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6 (2002) 405–432. | DOI | MR | Zbl
, and , Hybridization of mixed high-order methods on general meshes and application to the stokes equations. Comput. Methods Appl. Math. 15 (2015) 111–134. | DOI | MR | Zbl
S. Agmon, Lectures on elliptic boundary value problems. Vol. 2. American Mathematical Soc. (2010). | MR
, and , Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34 (1997) 828–852. | DOI | MR | Zbl
, , , and , Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Scientific Comput. 19 (1998) 404–425. | DOI | MR | Zbl
K. Aziz, Reservoir simulation grids: Opportunities and problems. J. Petroleum Technol. (1993) 658–663
S. Balay, M. Adams, J. Brown, P. Brune, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M. Knepley, L. McInnes, K. Rupp, B. Smith and H. Zhang, PETSc Web page, http://www.mcs.anl.gov/petsc (2014).
, and , Connection between finite volume and mixed finite element methods. RAIRO-Modél. Math. Anal. Numér. 30 (1996) 445–465. | DOI | Numdam | MR | Zbl
, , , , and , Cython: The best of both worlds. Comput. Sci. Eng. 13 (2011) 31–39. | DOI
, , , and , A gis-based open source pre-processor for georesources numerical modeling. Environ. Model. Softw. 62 (2014) 52–64. | DOI
S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods. Springer, 2nd edition (2002). | MR | Zbl
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991). | MR | Zbl
, and , Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217–235. | DOI | MR | Zbl
, and , Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005) 1872–1896. | DOI | MR | Zbl
, and , A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15 (2005) 1533–1551. | DOI | MR | Zbl
, and . Basic principles of mixed virtual element methods. ESAIM: M2AN 48 (2014) 1227–1240. | DOI | Numdam | MR | Zbl
and , Flux reconstruction and solution post-processing in mimetic finite difference methods. Comput. Methods Appl. Mech. Engrg. 197 (2008) 933–945. | DOI | MR | Zbl
, and , Unified hybridization of discontinuous galerkin, mixed, and continuous galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2009) 1319–1365. | DOI | MR | Zbl
, , , , and , Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. | DOI | MR | Zbl
L. Dalcin, petsc4py. https://code.google.com/p/petsc4py/ (2013).
, and , An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Math. 14 (2014) 461–472. | DOI | MR | Zbl
and , Superconvergence of mixed finite element methods on rectangular domains. Calcolo 26 (1989) 121–133. | DOI | MR | Zbl
, , and , A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl Sci. 20 (2010) 265–295. | DOI | MR | Zbl
, , and , Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci. 23 (2013) 2395–2432. | DOI | MR | Zbl
, Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6 (2002) 433–452. | DOI | MR | Zbl
and , Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2 (1998) 259–290. | DOI | MR | Zbl
, and , Superconvergence of the velocity along the gauss lines in mixed finite element methods. SIAM J. Numer. Anal. 28 (1991) 1015–1029. | DOI | MR | Zbl
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods. Vol. VII of Handbook of Numerical Analysis. North-Holland, Amsterdam (2000). | MR | Zbl
, , , , and , Meshvoro: A three-dimensional voronoi mesh building tool for the tough family of codes. Comput. Geosci. 70 (2014) 26–34. | DOI
et al., Parallel unstructured-solver methods for simulation of complex giant reservoirs. SPE J. 13 (2008) 440–446. | DOI
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Number 5 in Springer series in Computational Mathematics. Springer-Verlag (1986). | MR | Zbl
P. Grisvard, Elliptic problems in nonsmooth domains. Vol. 69. SIAM (2011). | MR | Zbl
, , and , M-adaptation in the mimetic finite difference method. Math. Models Methods Appl. Sci. 24 (2014) 1621–1663. | DOI | MR | Zbl
A. Henderson, The paraview guide: A parallel visualization application. kitware inc. Technical report (2007).
, and , A multipoint flux mixed finite element method on hexahedra. SIAM J. Numer. Anal. 48 (2010) 1281–1312. | DOI | MR | Zbl
E. Jones et al., SciPy: Open source scientific tools for Python (2001).
and , Convergence of multi-point flux approximation on general grids and media. Int. J. Numer. Anal. Model. 9 (2012) 584–606. | MR | Zbl
and , Convergence of multipoint flux approximations on quadrilateral grids. Numer. Methods Partial Differ. Eq. 22 (2006) 1438–1454. | DOI | MR | Zbl
and , Robust convergence of multi point flux approximation on rough grids. Numer. Math. 104 (2006) 317–337. | DOI | MR | Zbl
and , New mixed finite element method on polygonal and polyhedral meshes. Russian J. Numer. Anal. Math. Model. 18 (2003) 261–278. | DOI | MR | Zbl
, and , Local flux mimetic finite difference methods. Numer. Math. 112 (2009) 115–152. | DOI | MR | Zbl
, and , Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems. J. Comput. Phys. 230 (2011) 2620–2642. | DOI | MR | Zbl
, and , Mimetic finite difference method. Physics-compatible numerical methods. J. Comput. Phys. 257 (2014) 1163–1227. | DOI | MR | Zbl
, Fast and accurate computation of polyhedral mass properties. J. Graphics Tools 1 (1996) 31–50. | DOI
, and , Weak galerkin finite element methods on polytopal meshes. Int. J. Numer. Anal. Model. 12 (2015) 536–566. | MR
M. Nakata, A. Weiser and M. Wheeler, Some superconvergence results for mixed finite element methods for elliptic problems on rectangular domains. The Mathematics of Finite Elements and Applications V, edited by J.R. Whiteman. Academic Press, London (1985) 367–389. | MR | Zbl
and , Use of Voronoi grid in reservoir simluation. SPE Adv. Technol. Ser. 2 (1994) 69–77. | DOI
S. Panday, C.D. Langevin, R.G. Niswonger, M. Ibaraki and J.D. Hughes, Modflow-usg version 1: An unstructured grid version of MODFLOW for simulating groundwater flow and tightly coupled processes using a control volume finite-difference formulation. Technical report, US Geological Survey (2013).
P. Raviart and J. Thomas, A mixed finite element method for 2-nd order elliptic problems. In Mathematical Aspects of Finite Element Methods. Springer (1977) 292–315. | MR | Zbl
T. Russell and M. Wheeler, Finite element and finite difference methods for continuous flows in porous media. In Vol. 1 of Mathematics of Reservoir Simulation, Frontiers in Applied Mathematics, edited by R.E. Ewing. SIAM (1983). | MR | Zbl
C. Rycroft, Voro++: A three-dimensional voronoi cell library in c++ (2009).
Schlumberger Geoquest, Eclipse 100 reference manual. Schlumberger Geoquest (2001).
, and , The numpy array: a structure for efficient numerical computation. Comput. Sci. Eng. 13 (2011) 22–30. | DOI
and , Mixed finite element methods: implementation with one unknown per element, local flux expressions, positivity, polygonal meshes, and relations to other methods. Math. Models Methods Appl. Sci. 23 (2013) 803–838. | DOI | MR | Zbl
and , On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351–375. | DOI | MR | Zbl
, and , A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra. Numer. Math. 121 (2012) 165–204. | DOI | MR | Zbl
and , A multipoint flux mixed finite element method. SIAM J. Numer. Anal. 44 (2006) 2082–2106. | DOI | MR | Zbl
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