Symmetric, unconditionnaly coercive schemes for the discretization of heterogeneous and anisotropic diffusion problems on general, possibly nonconforming meshes are developed and studied. These schemes are a further generalization of the Hybrid Mixed Method, which allows to use a general class of consistent gradients to construct them. While the schemes are in principle hybrid, many discrete gradients or the use of correct interpolation allow to eliminate the additional face unknowns. Convergence of the approximate solutions to minimal regularity solutions is proved for general tensors and meshes. Error estimates are derived under classical regularity assumptions. Numerical results illustrate the performance of the schemes.
DOI : 10.1051/m2an/2015005
Mots-clés : Heterogeneous diffusion problems, cell-centered methods, hybrid finite volumes, general meshes
@article{M2AN_2015__49_4_1063_0,
author = {Coatl\'even, Julien},
title = {Semi hybrid method for heterogeneous and anisotropic diffusion problems on general meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1063--1084},
publisher = {EDP-Sciences},
volume = {49},
number = {4},
year = {2015},
doi = {10.1051/m2an/2015005},
mrnumber = {3371904},
zbl = {1327.65206},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2015005/}
}
TY - JOUR AU - Coatléven, Julien TI - Semi hybrid method for heterogeneous and anisotropic diffusion problems on general meshes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1063 EP - 1084 VL - 49 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2015005/ DO - 10.1051/m2an/2015005 LA - en ID - M2AN_2015__49_4_1063_0 ER -
%0 Journal Article %A Coatléven, Julien %T Semi hybrid method for heterogeneous and anisotropic diffusion problems on general meshes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1063-1084 %V 49 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2015005/ %R 10.1051/m2an/2015005 %G en %F M2AN_2015__49_4_1063_0
Coatléven, Julien. Semi hybrid method for heterogeneous and anisotropic diffusion problems on general meshes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 1063-1084. doi : 10.1051/m2an/2015005. http://archive.numdam.org/articles/10.1051/m2an/2015005/
, , and , Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127 (1996) 2–14. | DOI | Zbl
, , and , Discretization on unstructured grids for inhomogeneous, anisotropic media part i: Derivation of the methods. SIAM J. Sci. Comput. 19 (1998a) 1700–1716. | DOI | MR | Zbl
, , and , Discretization on unstructured grids for inhomogeneous, anisotropic media. part ii: Discussion and numerical results. SIAM J. Sci. Comput. 19 (1998b) 1717–1736. | DOI | MR | Zbl
and , Convergence of finite volume mpfa o type schemes for heterogeenous anisotropic diffusion problems on general meshes. C.R. Acad. Paris Ser. I 346 (2008). | MR | Zbl
, and , The g method for heterogeneous anisotropic diffusion on general meshes. ESAIM: M2AN 11 (2010) 597–625. | DOI | Numdam | MR | Zbl
, , and , An abstract analysis framework for nonconforming approximations of anistropic heterogeneous diffusion problems. IJFV International Journal On Finite Volumes 7 (2010). | MR
L. Agélas, D.A. Di Pietro and R. Masson, A symmetric and coercive finite volume scheme for multiphase porous media flow problems with applications in the oil industry. In Finite volume for Complex Applications V. Edited by R. Eymard ans J.-M. Hérard. Wiley (2008) 35–51. | MR
, , , , and , Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. | DOI | MR | Zbl
L. Beirao da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method for Elliptic Problems. Springer (2014). | MR | Zbl
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | MR | Zbl
, and , Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005a) 1872–1896. | DOI | MR | Zbl
, and , A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15 (2005b) 1533–1551. | DOI | MR | Zbl
, Finite volume schemes for diffusion equations: introduction to and review of modern methods. Special edition “P.D.E. Discretizations on Polygonal Meshes”. M3AS 24 (2014) 1575–1619. | MR | Zbl
and , A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105 (2006) 35–71. | DOI | MR | Zbl
, , and , A unified approach to mimetic finite differences, hybrid finite volume and mixed finite volume methods. IMA J. Numer. Anal. 31 (2011) 1357–1401. | Zbl
, , and , Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. M3AS 23 (2013) 2395–2432. | MR | Zbl
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, In Techniques of scientific computiing, Part III. Handb. Numer. Anal. Edited by P.G. Ciarlet and J.-L. Lions. North-Holland, Amsterdam (2000) 713–1020. | MR | Zbl
, and , A new finite volume scheme for anisotropic diffusion problems on general grids: convergence analysis. C.R. Math. Acad. Sci. Paris 344 (2007a) 403–406. | DOI | MR | Zbl
, and , Discretisation of heterogeneous and anisotropic diffusion problems on general nonconforming meshes sushi: a scheme using stabilisation and hybrid interfaces. IMA J. Numer. Anal. 30 (2010) 1009–1043x. | DOI | MR | Zbl
and , A new colocated finite volume scheme for the incompressible navier-stokes equations on general non matching grids. C.R. Math. Acad. Sci. Paris 344 (2007b) 659–662. | DOI | MR | Zbl
, and , Small stencil 3d schemes for diffusive flows in porous media. ESAIM: M2AN 46 (2012) 265–290. | DOI | Numdam | MR | Zbl
, Cell centered galerkin methods. C.R. Acad. Sci. Paris Ser. I 348 (2010) 31–34. | DOI | MR | Zbl
, Cell centered galerkin methods for diffusive problems. ESAIM: M2AN 46 (2012) 111–144. | DOI | Numdam | MR | Zbl
, Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes. ESAIM: M2AN 40 (2006) 367–391. | DOI | Numdam | MR | Zbl
and , From face to element unknowns by local static condensation with application to nonconforming finite elements. Comput. Methods Appl. Mech. Eng. 253 (2013a) 517–529. | DOI | MR | Zbl
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 (2013b) 803–838. | DOI | MR | Zbl
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