Discretization on Unstructured Grids for Inhomogeneous, Anisotropic Media. Part I: Derivation of the Methods

Aavatsmark, Ivar; Barkve, T.; Bøe, Ø.

doi:10.1137/s1064827595293582

Cited by 315 publications

(297 citation statements)

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“…We refer to Eymard, Gallouët and Herbin [52] for the description and comparison of these and related (e.g., mimetic finite difference) schemes. Finally, let us also mention the schemes of Aavatsmark et al (see, e.g., [1,2]), that are in a sense intermediate. The gradient reconstruction used in [1,2] also involves additional edge unknowns, which are eliminated by solving, locally, an algebraic system of equations.…”

Section: On the Choice Of Fv Scheme And Various Generalizationsmentioning

confidence: 99%

“…The initial data u 0 : Ω → R are assumed to be a bounded measurable function, i.e., u 0 ∈ L ∞ (Ω), while the source S : Q → R is assumed to be a measurable function for which S(t, ·) ∈ L ∞ (Ω) for a.e. t ∈ (0, T ) and T 0 S(t, ·) L ∞ (Ω) dt < ∞; we abusively denote it by (2) S ∈ L 1 (0, T ; L ∞ (Ω)).…”

Section: Introductionmentioning

confidence: 99%

“…We refer to [48,31,3,44,45,61,57,68,79,49,67,12,10,11,42] and references therein for different convergence results and numerical experiments. For related works on linear elliptic problems, see [2,1,57,41,23,58,50,51,53,52] and the discussion in Section 8. Alternative numerical approaches have also been investigated; here we only mention finite element schemes (see [36,16] and references therein), kinetic schemes (see [14,22,55] and references therein) and operator splitting schemes (see [43]).…”

Section: Introductionmentioning

confidence: 99%

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Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Andreïanov¹,

Boyer²,

Hubert³

2006

Numerical Methods Partial

129

275

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Abstract. We consider a class of doubly nonlinear degenerate hyperbolicparabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [41]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" L ∞ weak-⋆ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, . . . ). Our results cover the case of non-Lipschitz nonlinearities.

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Section: On the Choice Of Fv Scheme And Various Generalizationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Andreïanov¹,

Boyer²,

Hubert³

2006

Numerical Methods Partial

129

275

View full text Add to dashboard Cite

show abstract

“…Eymard et al [36], Aavatsmark et al [1], or Droniou and Eymard [32], mimetic finite difference; cf. Brezzi et al [21], covolume; cf.…”

Section: Introductionmentioning

confidence: 99%

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Vohralı́k

2010

Math. Comp.

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Abstract. We derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second-order elliptic problems. It is based on the classical primal weak formulation, the postprocessing of the potential proposed in [T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for secondorder elliptic problems, Math. Comp. 64 (1995), 943-972], and the discrete Friedrichs inequality. Our analysis in particular avoids any explicit use of the uniform discrete inf-sup condition and in a straightforward manner and under minimal necessary assumptions, known convergence and superconvergence results are recovered. The same framework then turns out to lead to optimal a posteriori energy error bounds. In particular, estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived. They give a guaranteed and fully computable upper bound on the energy error, represent error local lower bounds, and are robust under some conditions on the diffusion-dispersion tensor. They are thus suitable for both overall error control and adaptive mesh refinement. Moreover, the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method. We finally prove that in parallel and simultaneously in converse to Galerkin finite element methods, under some circumstances, the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the H 1 0 (Ω) space and also establish several links between mixed finite element approximations and some generalized weak solutions.

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“…CVMFEM are closely related to the Raviart-Thomas mixed finite element methods (MFEM) [26,7,27], cell-centered finite difference (CCFD) methods [28,29,4], mimetic finite difference (MFD) methods [5,21,6], and multipoint flux approximation (MPFA) methods [1,17]. Some of these relationships are explored in detail in [22].…”

Section: Introductionmentioning

confidence: 99%

Superconvergence for Control‐Volume Mixed Finite Element Methods on Rectangular Grids

Russell

Wheeler²,

Yotov

2007

SIAM J. Numer. Anal.

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We consider control-volume mixed finite element methods for the approximation of second-order elliptic problems on rectangular grids. These methods associate control volumes (covolumes) with the vector variable as well as the scalar, obtaining local algebraic representation of the vector equation (e.g., Darcy's law) as well as the scalar equation (e.g., conservation of mass). We establish O(h 2 ) superconvergence for both the scalar variable in a discrete L 2 -norm and the vector variable in a discrete H(div)-norm. The analysis exploits a relationship between control-volume mixed finite element methods and the lowest order Raviart-Thomas mixed finite element methods.

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Discretization on Unstructured Grids for Inhomogeneous, Anisotropic Media. Part I: Derivation of the Methods

Cited by 315 publications

References 10 publications

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Superconvergence for Control‐Volume Mixed Finite Element Methods on Rectangular Grids

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