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## Local flux mimetic finite difference methods (2005)

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Citations: | 23 - 6 self |

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2515 |
The Finite Element Method for Elliptic Problems
- Ciarlet
- 1978
(Show Context)
Citation Context ... properties for these quadratures are required to prove the optimal convergence rates. Fourth, we derive a discrete formula for the other operator. Let Ωh be a conforming shape-regular partition (see =-=[8]-=-) of the computational domain into polygonal elements. Let Ω h = max hE, E∈Ωh where hE is the diameter of element E. We assume that each vertex of E is shared by exactly d edges (faces in 3D) of that ... |

1667 |
Mixed and hybrid finite element methods
- Brezzi, Fortin
- 1991
(Show Context)
Citation Context ... . The vector transformation is known as the Piola transformation. It is designed to preserve the normal components of the velocity vectors on the edges (faces) and satisfies the important properties =-=[15]-=- � � div�v w dx = divˆ � � �v ˆw dˆx and �v · �ne w ds = ˆ�v · ˆ �nê ˆw dˆs. (A.2) E Ê For the standard change of variables, we have � � w ds = ˆw Je dˆs, Je = JE�DF −T E ˆ �nê�. (A.3) e ê The above r... |

1516 | The mathematical theory of finite element methods (Vol - Brenner, Scott - 2008 |

916 |
Non-Homogeneous Boundary Value Problems and Applications. Vol
- Lions, Magenes
- 1972
(Show Context)
Citation Context ...) in Ω, ˜e ϕ = 0 on ∂Ω, where R(p I − ph) is the piecewise constant function equal to (p I − ph)E on each element E. The regularity assumption implies �ϕ� H 2 (Ω) ≤ C�R(p I − ph)� L 2 (Ω); (3.26) see =-=[11, 17]-=- for sufficient conditions. Let � ψ = −K grad ϕ. Let (·, ·) denote the L 2 inner product over Ω. Using Lemma 3.3 and (2.18), we get �R(p I − ph)� 2 L 2 (Ω) = (R(pI − ph), div Π � ψ) where = (p, div Π ... |

767 | equations. Theory and numerical analysis. Revised edition. With an appendix by F. Thomasset - Temam, Navier-Stokes - 1979 |

731 |
Elliptic problems in nonsmooth domains
- Grisvard
- 1985
(Show Context)
Citation Context ...) in Ω, ˜e ϕ = 0 on ∂Ω, where R(p I − ph) is the piecewise constant function equal to (p I − ph)E on each element E. The regularity assumption implies �ϕ� H 2 (Ω) ≤ C�R(p I − ph)� L 2 (Ω); (3.26) see =-=[11, 17]-=- for sufficient conditions. Let � ψ = −K grad ϕ. Let (·, ·) denote the L 2 inner product over Ω. Using Lemma 3.3 and (2.18), we get �R(p I − ph)� 2 L 2 (Ω) = (R(pI − ph), div Π � ψ) where = (p, div Π ... |

520 |
An introduction to the mathematical theory of the Navier - Stokes equations, I
- Galdi
- 1994
(Show Context)
Citation Context ... − q0 in Ω, �v1 = 0 on ∂Ω, Ω div �v2 = q0 in Ω, �v2 = �g on ∂Ω, where �g ∈ (H1/2 (∂Ω)) d and satisfies the compatibility condition � �g · �n ds = q0|Ω|. The above problems are known to have solutions =-=[10]-=- satisfying ∂Ω ��v1� (H1 (Ω)) d ≤ C�q�L2 (Ω) and ��v2� (H1 (Ω)) d ≤ C � � �q0�L2 (Ω) + ��g� (H1/2 (∂Ω)) d . We choose �g = |Ω|q0φ�n, where φ is a smooth function with support contained within one side... |

280 |
An interior penalty finite element method with discontinuous elements
- Arnold
- 1982
(Show Context)
Citation Context ...7) In particular, there exists a linear function p 1 E such that �p − p 1 E�L 2 (E) ≤ C h 2 E �p�H 2 (E), �p − p 1 E�H 1 (E) ≤ C hE �p�H 2 (E). (3.8) For the error on the edges (faces in 3D), we have =-=[3]-=- �χ� 2 L2 (˜e) ≤ C � h −1 E �χ�2 L 2 (E) + hE |χ| 2 H 1 (E) � , ∀χ ∈ H 1 (E), (3.9) where ˜e is any edge (face) of E. The constant C in (3.8) and (3.9) depends only on the shape-regularity constants o... |

265 |
A Mixed Finite Element Method for Second Order Elliptic Problems
- RAVIART, THOMAS
- 1977
(Show Context)
Citation Context ... can be reduced to a cell-centered scheme with a local stencil. A close relationship between the MFD method and the mixed finite element (MFE) method with the lowest order Raviart-Thomas elements RT0 =-=[42]-=- has been established in [9]. There, it is shown that the spaces of discrete mimetic degrees of freedom on triangles and quadrilaterals are isomorphic to the RT0 spaces; moreover, the MFD method can b... |

235 |
Mixed and nonconforming finite element methods : Implementation, postprocessing and error estimates
- Arnold, Brezzi
- 1985
(Show Context)
Citation Context ...is applies to more general polyhedral meshes. The MFE method, like the MFD method, leads to a saddle-point problem. Several approaches have been proposed to handle this issue, including hybridization =-=[7]-=- and reduction to cell-centered finite differences (CCFD) [43, 46, 5, 4, 8, 39]. These methods, however, either lead to a more expensive face-centered stencil [7], or limited to diagonal tensor coeffi... |

157 |
Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace
- Adams
- 1975
(Show Context)
Citation Context ...rem [45]. For �v ∈ V(D), define γn�v by � � 〈γn�v,φ〉∂D := div�v w dx + �v · ∇w dx ∀φ ∈ W 1/˜s,˜s′ (∂D), D D where w ∈ W 1,˜s′ (D) is a continuous extension of φ in D. By the Sobolev imbedding theorem =-=[3]-=-, w ∈ L2 (D), therefore |〈γn�v,φ〉∂D| ≤ C(�div�v� L 2 (D) + ��v� (L ˜s (D)) d)�w� W 1,˜s ′ (D) ≤ C(�div�v� L 2 (D) + ��v� (L ˜s (D)) d)�φ� W 1/˜s,˜s ′ (∂D) , hence the map γn : V(D) → (W 1/˜s,˜s′ (∂D))... |

125 |
The finite volume method, Handbook Numer. Anal
- Eymard, Gallouët, et al.
- 2000
(Show Context)
Citation Context ...imetic spaces without the use of finite element polynomial extensions, except in the pressure superconvergence proof. In terms of computational cost, our method is comparable to finite volume methods =-=[22]-=-. However, the latter are either limited to diagonal tensor coefficients, or require certain orthogonality properties of the grid [23], or need to be augmented with face-centered pressures [24], which... |

112 |
Finite element and finite difference methods for continuous flows in porous media
- Russell, Wheeler
- 1983
(Show Context)
Citation Context ... like the MFD method, leads to a saddle-point problem. Several approaches have been proposed to handle this issue, including hybridization [7] and reduction to cell-centered finite differences (CCFD) =-=[43, 46, 5, 4, 8, 39]-=-. These methods, however, either lead to a more expensive face-centered stencil [7], or limited to diagonal tensor coefficients [43, 46, 8, 39], or exhibit deterioration of convergence for discontinuo... |

96 | Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences
- Arbogast, Wheeler, et al.
(Show Context)
Citation Context ... like the MFD method, leads to a saddle-point problem. Several approaches have been proposed to handle this issue, including hybridization [7] and reduction to cell-centered finite differences (CCFD) =-=[43, 46, 5, 4, 8, 39]-=-. These methods, however, either lead to a more expensive face-centered stencil [7], or limited to diagonal tensor coefficients [43, 46, 8, 39], or exhibit deterioration of convergence for discontinuo... |

86 | M.: Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes
- Brezzi, Lipnikov, et al.
(Show Context)
Citation Context ...sfying (2.10), we have [�u I − uh, v]X = [(K grad p) I − (K grad p 1 ) I , v]X + [(K grad p 1 ) I − (K grad p 1 ) I , v]X + [(K grad p 1 ) I , v]X ≡ I1 + I2 + I3. Terms similar to I1 and I2 appear in =-=[6]-=-. Using the Cauchy-Schwarz inequality, we bound I1 as |I1| ≤ |(K grad p − K grad p 1 ) I |X |v |X � � � � � 1 I e 2 ≤ α1 ((K grad p − K grad p ) ) E |E| = � α1 E∈Ωh e∈∂E � � E∈Ωh e∈∂E ≤ Ch�p�H 2 (Ω) |... |

82 |
Discretization on unstructured grids for inhomogeneous, anisotropic media. Part II: Discussion and numerical results
- Aavatsmark, Barkve, et al.
- 1998
(Show Context)
Citation Context ...vertex (corner). To approximate the pressure, the method uses one degree of freedom per element. These choices are similar to the degrees of freedom in the multipoint flux approximation (MPFA) method =-=[2, 1, 9]-=-. A specially chosen flux inner product couples only the flux degrees of freedom associated with each mesh vertex and allows for local flux elimination, reducing the method to a symmetric cell-centere... |

67 |
Two families of mixed elements for second order elliptic problems
- Brezzi, Jr, et al.
- 1985
(Show Context)
Citation Context ...onal interpolation operators. Let Vh be the lowest order Brezzi-Douglas-Marini BDM1 mixed finite element space on Ωh, consisting of piecewise linear vector functions with continuous normal components =-=[5]-=-. For any �v ∈ (L s (Ω)) d , s > 2, let Π�v ∈ Vh be its finite element interpolant satisfying for every element edge (face in 3D) ˜e ∈ ∂E � (Π�v − �v) · �nE p1 ds = 0 for every linear function p1. (3.... |

67 | A finite volume scheme for anisotropic diffusion problems
- Eymard, Gallouët, et al.
(Show Context)
Citation Context ...ethods [22]. However, the latter are either limited to diagonal tensor coefficients, or require certain orthogonality properties of the grid [23], or need to be augmented with face-centered pressures =-=[24]-=-, which increases their cost. The paper outline is as follows. The new MFD method is developed in Section 2. In Section 3, we prove convergence estimates for the pressure and the velocity variables un... |

62 | S.: The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials
- Hyman, Shashkov, et al.
- 1997
(Show Context)
Citation Context ...employed for solving problems of continuum mechanics [19], electromagnetics [13], gas dynamics [7], and linear diffusion on polygonal and polyhedral meshes in both the Cartesian and polar coordinates =-=[14, 20, 18]-=-. The MFD method mimics essential properties of the continuum equations, such as conservation laws, solution symmetries, and the fundamental identities and theorems of vector and tensor calculus. For ... |

59 | A family of mimetic finite difference methods on polygonal and polyhedral meshes
- Brezzi, Lipnikov, et al.
- 2005
(Show Context)
Citation Context ... in [9, 11, 10] to establish convergence and superconvergence for the MFD approximations on simplicial and quadrilateral elements. An alternative approach for analyzing the MFD method is developed in =-=[16, 17]-=-, where the error in appropriate discrete norms is estimated. The main advantage of this approach is that the analysis applies to more general polyhedral meshes. The MFE method, like the MFD method, l... |

56 |
Navier-Stokes equations. AMS Chelsea Publishing
- Temam
- 2001
(Show Context)
Citation Context ...n. Moreover, the Green’s formula � � � γn�v w ds = div�v w dx + �v · ∇w dx ∂D holds for all �v ∈ V(D) and w ∈ W 1,˜s′ (D). D Proof. The proof is a generalization of the classical normal trace theorem =-=[45]-=-. For �v ∈ V(D), define γn�v by � � 〈γn�v,φ〉∂D := div�v w dx + �v · ∇w dx ∀φ ∈ W 1/˜s,˜s′ (∂D), D D where w ∈ W 1,˜s′ (D) is a continuous extension of φ in D. By the Sobolev imbedding theorem [3], w ∈... |

51 | I.: Enhanced cell-centered finite differences for elliptic equations on general geometry
- Arbogast, Dawson, et al.
- 1998
(Show Context)
Citation Context ... like the MFD method, leads to a saddle-point problem. Several approaches have been proposed to handle this issue, including hybridization [7] and reduction to cell-centered finite differences (CCFD) =-=[43, 46, 5, 4, 8, 39]-=-. These methods, however, either lead to a more expensive face-centered stencil [7], or limited to diagonal tensor coefficients [43, 46, 8, 39], or exhibit deterioration of convergence for discontinuo... |

50 |
Algebraic multigrid (AMG): experiences and comparisons
- Stüben
- 1983
(Show Context)
Citation Context ...symmetric positive definite matrix. This problem is solved with the preconditioned conjugate gradient (PCG) method. In the numerical experiments, we used one V-cycle of the algebraic multigrid method =-=[21]-=- as a preconditioner. The stopping criterion for the PCG method is the relative decrease in the residual norm by a factor of 10 −12 . Let us consider the 2D problem (2.2) in the unit square with the k... |

45 |
An introduction to multipoint flux approximations for quadrilateral grids
- Aavatsmark
(Show Context)
Citation Context ...vertex (corner). To approximate the pressure, the method uses one degree of freedom per element. These choices are similar to the degrees of freedom in the multipoint flux approximation (MPFA) method =-=[2, 1, 9]-=-. A specially chosen flux inner product couples only the flux degrees of freedom associated with each mesh vertex and allows for local flux elimination, reducing the method to a symmetric cell-centere... |

44 |
Finite volume discretization with imposed flux continuity for the general tensor pressure equation
- Edwards, Rogers
- 1998
(Show Context)
Citation Context ...vertex (corner). To approximate the pressure, the method uses one degree of freedom per element. These choices are similar to the degrees of freedom in the multipoint flux approximation (MPFA) method =-=[2, 1, 9]-=-. A specially chosen flux inner product couples only the flux degrees of freedom associated with each mesh vertex and allows for local flux elimination, reducing the method to a symmetric cell-centere... |

39 |
Connection between finite volume and mixed finite element methods
- Baranger, Maitre, et al.
- 1996
(Show Context)
Citation Context |

35 | M.: A tensor artificial viscosity using a mimetic finite difference algorithm
- Campbell, Shashkov
- 2001
(Show Context)
Citation Context ...N06, 65N12, 65N15, 65N30 1 Introduction The mimetic finite difference (MFD) method has been successfully employed for solving problems of continuum mechanics [19], electromagnetics [13], gas dynamics =-=[7]-=-, and linear diffusion on polygonal and polyhedral meshes in both the Cartesian and polar coordinates [14, 20, 18]. The MFD method mimics essential properties of the continuum equations, such as conse... |

31 | A local support-operators diffusion discretization scheme for quadrilateral r-z meshes
- Morel, Roberts, et al.
- 1998
(Show Context)
Citation Context ...employed for solving problems of continuum mechanics [19], electromagnetics [13], gas dynamics [7], and linear diffusion on polygonal and polyhedral meshes in both the Cartesian and polar coordinates =-=[14, 20, 18]-=-. The MFD method mimics essential properties of the continuum equations, such as conservation laws, solution symmetries, and the fundamental identities and theorems of vector and tensor calculus. For ... |

28 | A multipoint flux mixed finite element method
- Wheeler, Yotov
(Show Context)
Citation Context ...convergence for both the flux and the pressure variables, as well as superconvergence of the pressure in discrete L2 norms. Our analysis can be extended to smooth quadrilateral meshes. Recent results =-=[15, 16, 22]-=- provide analysis for the MPFA method and some related mixed finite element methods by employing finite element techniques. Our approach is based on estimating the errors directly in the norms of the ... |

28 | M.: Convergence of mimetic finite difference discretizations of the diffusion equation
- Berndt, Lipnikov, et al.
- 2001
(Show Context)
Citation Context ...tered scheme with a local stencil. A close relationship between the MFD method and the mixed finite element (MFE) method with the lowest order Raviart-Thomas elements RT0 [42] has been established in =-=[9]-=-. There, it is shown that the spaces of discrete mimetic degrees of freedom on triangles and quadrilaterals are isomorphic to the RT0 spaces; moreover, the MFD method can be viewed as a MFE method wit... |

28 |
Domain decomposition and iterative refinement methods for mixed finite element discretizations of elliptic problems
- Mathew
- 1989
(Show Context)
Citation Context ...ve �φ − φ 1 E� 2 L2 (˜e) ≤ C h1+2q E |φ| 2 H1+q (E) , 0 ≤ q ≤ 1. (3.6) The estimate also holds for any facet e of E. Let ˜V(E) = {�v: �v ∈ (H ˜q (E)) d , 0 < ˜q ≤ 1, div�v ∈ L 2 (E)}. It was shown in =-=[38]-=- that the map �v · �n : ˜ V(E) → H ˜q−1/2 (˜e), 0 < ˜q ≤ 1, is continuous, where H ˜q−1/2 (˜e) = (H 1/2−˜q (˜e)) ∗ . The following result is proved in Appendix A using a scaling argument. Lemma 3.1 Le... |

27 | Mimetic discretizations for Maxwell’s equations and the equations of magnetic diffusion, Prog. Electromagnetic Res - Hyman, Shashkov |

24 | P.: A discrete operator calculus for finite difference approximations
- Margolin, Shashkov, et al.
- 2000
(Show Context)
Citation Context ...r estimates AMS Subject Classification: 65N06, 65N12, 65N15, 65N30 1 Introduction The mimetic finite difference (MFD) method has been successfully employed for solving problems of continuum mechanics =-=[19]-=-, electromagnetics [13], gas dynamics [7], and linear diffusion on polygonal and polyhedral meshes in both the Cartesian and polar coordinates [14, 20, 18]. The MFD method mimics essential properties ... |

23 | The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes
- Lipnikov, Shashkov, et al.
(Show Context)
Citation Context ...employed for solving problems of continuum mechanics [19], electromagnetics [13], gas dynamics [7], and linear diffusion on polygonal and polyhedral meshes in both the Cartesian and polar coordinates =-=[14, 20, 18]-=-. The MFD method mimics essential properties of the continuum equations, such as conservation laws, solution symmetries, and the fundamental identities and theorems of vector and tensor calculus. For ... |

21 |
A cell-centered finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension
- Eymard, Gallouët, et al.
(Show Context)
Citation Context ...utational cost, our method is comparable to finite volume methods [22]. However, the latter are either limited to diagonal tensor coefficients, or require certain orthogonality properties of the grid =-=[23]-=-, or need to be augmented with face-centered pressures [24], which increases their cost. The paper outline is as follows. The new MFD method is developed in Section 2. In Section 3, we prove convergen... |

20 | The approximation of boundary conditions for mimetic finite difference methods
- Hyman, Shashkov
- 1998
(Show Context)
Citation Context ...tem of two first order equations �u = −K grad p in Ω, (2.2) div �u = f in Ω, subject to appropriate boundary conditions. For simplicity, we consider the homogeneous Dirichlet boundary conditions (see =-=[12]-=- for more general boundary conditions): p = 0 on ∂Ω. (2.3) We consider a polygonal domain Ω ⊂ R d , d = 2 or 3, with boundary ∂Ω and outward unit normal �n. The coefficient K is a symmetric and unifor... |

18 | Convergence of multipoint flux approximations on quadrilateral grids
- KLAUSEN, WINTHER
- 2006
(Show Context)
Citation Context ...convergence for both the flux and the pressure variables, as well as superconvergence of the pressure in discrete L2 norms. Our analysis can be extended to smooth quadrilateral meshes. Recent results =-=[15, 16, 22]-=- provide analysis for the MPFA method and some related mixed finite element methods by employing finite element techniques. Our approach is based on estimating the errors directly in the norms of the ... |

18 | I.: Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals
- Berndt, Lipnikov, et al.
(Show Context)
Citation Context ...adrilaterals are isomorphic to the RT0 spaces; moreover, the MFD method can be viewed as a MFE method with a quadrature rule for calculating the velocity mass matrix. This relationship is explored in =-=[9, 11, 10]-=- to establish convergence and superconvergence for the MFD approximations on simplicial and quadrilateral elements. An alternative approach for analyzing the MFD method is developed in [16, 17], where... |

18 |
New mixed finite element method on polygonal and polyhedral meshes
- Kuznetsov, Repin
(Show Context)
Citation Context ... from Theorem 3.3 to polygonal and polyhedral elements requires verifying assumptions A6 and A7. One could construct appropriate interpolation operators on such elements by extending the results from =-=[32, 33]-=- on piecewise Raviart-Thomas spaces to piecewise BDM1 spaces. 5 Numerical experiments In this section, we present results of numerical experiments using quadrature rules defined in (4.2). As we mentio... |

17 | Robust convergence of multipoint flux approximation on rough grids. Numerische Mathematik 2006 - RA, Winther |

16 | Relationships among some locally conservative discretization methods wich handle discontinuous coefficients
- Klausen, Russell
(Show Context)
Citation Context ...convergence for both the flux and the pressure variables, as well as superconvergence of the pressure in discrete L2 norms. Our analysis can be extended to smooth quadrilateral meshes. Recent results =-=[15, 16, 22]-=- provide analysis for the MPFA method and some related mixed finite element methods by employing finite element techniques. Our approach is based on estimating the errors directly in the norms of the ... |

14 |
Discretisation and multigrid solution of elliptic equations with mixed derivatives terms and strongly discontinuous coefficients
- Crumpton, Shaw, et al.
- 1995
(Show Context)
Citation Context ...t works [29, 30, 47] establish relationships between the MFE method and the multipoint flux approximation (MPFA) method introduced by the petroleum reservoir simulation community [2, 1, 21], see also =-=[20, 34, 12]-=- for closely related methods. The MPFA method, which is formulated as a finite volume method, utilizes sub-edge fluxes and reduces to a cell-centered pressure scheme through local flux elimination. Pa... |

13 |
A cell-centered diffusion scheme on two-dimensional unstructured meshes
- Breil, Maire
(Show Context)
Citation Context ...t works [29, 30, 47] establish relationships between the MFE method and the multipoint flux approximation (MPFA) method introduced by the petroleum reservoir simulation community [2, 1, 21], see also =-=[20, 34, 12]-=- for closely related methods. The MPFA method, which is formulated as a finite volume method, utilizes sub-edge fluxes and reduces to a cell-centered pressure scheme through local flux elimination. Pa... |

13 |
Finite difference schemes on triangular cell-centered grids with local refinement
- Vassilevski, Petrova, et al.
- 1992
(Show Context)
Citation Context |

13 | Convergence analysis and error estimates for mixed finite element method on distorted meshes - Kuznetsov, Repin - 2005 |

8 |
Schéma volumes finis pour des opérateurs de diffusion fortement anisotropes sur des maillages non structurés
- Potier
(Show Context)
Citation Context ...t works [29, 30, 47] establish relationships between the MFE method and the multipoint flux approximation (MPFA) method introduced by the petroleum reservoir simulation community [2, 1, 21], see also =-=[20, 34, 12]-=- for closely related methods. The MPFA method, which is formulated as a finite volume method, utilizes sub-edge fluxes and reduces to a cell-centered pressure scheme through local flux elimination. Pa... |

6 |
On Some Mixed Finite Element Methods with Numerical Integration
- MICHELETTI, SACCO, et al.
(Show Context)
Citation Context |

4 | A mortar mimetic finite difference method on non-matching grids
- Berndt, Lipnikov, et al.
(Show Context)
Citation Context ...adrilaterals are isomorphic to the RT0 spaces; moreover, the MFD method can be viewed as a MFE method with a quadrature rule for calculating the velocity mass matrix. This relationship is explored in =-=[9, 11, 10]-=- to establish convergence and superconvergence for the MFD approximations on simplicial and quadrilateral elements. An alternative approach for analyzing the MFD method is developed in [16, 17], where... |

2 |
convergence of multi point flux approximation on rough grids
- Robust
(Show Context)
Citation Context ...heme through local flux elimination. Papers [30] and [47] study the convergence properties of the MPFA method and related MFE methods with broken RT0 and BDM1 [14] spaces, respectively. More recently =-=[31]-=- analyzes the convergence of a non-symmetric MPFA method on general quadrilateral grids. In this paper, we employ a MPFA-type construction and analysis inspired by [16] to develop new cell-centered di... |

2 |
Mimetic preconditioners for mixed discretizations of the diffusion equation
- MOREL, MOULTON, et al.
- 2001
(Show Context)
Citation Context ...locity and the pressure variables, as well as second-order superconvergence for the pressure variable in discrete L 2 norms. For simplicial meshes, we employ a symmetric quadrature rule introduced in =-=[40]-=- and similar to the vector inner product used in [47], and prove that the constructive assumptions hold. These results can be extended to smooth quadrilateral and hexahedral meshes. For general polyhe... |