### Citations

2575 |
The Finite Element Method for Elliptic Problems
- Ciarlet
- 1978
(Show Context)
Citation Context ...tic size h. The elements can be triangles or quadrilaterals in 2D, tetrahedra or hexahedra in 3D. Let us denote the family of partitions by Th and assume that they are shape-regular and quasi-uniform =-=[12]-=-. This is a standard assumption in finite element analysis. It was needed in the analysis of the MEMFE method for Darcy flow in [21, 44, 47]. We utilize several auxiliary results from these papers in ... |

1697 | Mixed and Hybrid Finite Element Methods - Brezzi, Fortin - 1991 |

1544 |
The Mathematical Theory of Finite Element Methods.
- Brenner, Scott
- 1994
(Show Context)
Citation Context ...ϕ − Qhϕ)t‖ h‖ϕt‖1. (2.60) Comput Geosci (2014) 18:57–75 63 Fig. 1 Interactions of the velocity degrees of freedom in the MFMFE method In the analysis, we will use the following well-known estimates =-=[7]-=-. There exists s1 ∈ P1(E) such that ‖p − s1‖j,E h2−j‖p‖2,E, j = 0, 1, (2.61) and ‖p − s1‖E h‖p‖1,E. (2.62) We also have [12] ‖K − KE‖E h‖K‖1,E. (2.63) Lemma 2.3 ([47]) For all q ∈ (H 1(E))d , ‖ ... |

463 |
General theory of threedimensional consolidation.
- Biot
- 1941
(Show Context)
Citation Context ...ng, wellbore collapse, sand production, and fault activation. The mathematical model for the coupled fluid-solid system used in this paper is the classical Biot consolidation system in poroelasticity =-=[6, 42]-=- under a quasi-static assumption. The system consists of an equilibrium equation for the solid and a mass balance equation for the fluid. The fluid pressure contributes to the total stress of the soli... |

312 |
A mixed finite element method for 2nd order elliptic problems.
- Raviart, Thomas
- 1977
(Show Context)
Citation Context ...‖1,E. (2.63) Lemma 2.3 ([47]) For all q ∈ (H 1(E))d , ‖ q‖E ‖q‖E + h|q|1,E. (2.64) In the analysis, we will require a similar projection operator onto the lowest order Raviart-Thomas velocity space =-=[34, 39]-=-. The RT0 spaces are defined on the unit cube as ẐR(Ê) = ⎛ ⎝ α1 + β1x̂ α2 + β2ŷ α3 + β3ẑ ⎞ ⎠ , ŴR(Ê) = P0(Ê), (2.65) and on the unit square as ẐR(Ê) = ( α1 + β1x̂ α2 + β2ŷ ) , ŴR(Ê) = P0(E... |

267 |
Mixed finite elements in R3,”
- Nedelec
- 1980
(Show Context)
Citation Context ...‖1,E. (2.63) Lemma 2.3 ([47]) For all q ∈ (H 1(E))d , ‖ q‖E ‖q‖E + h|q|1,E. (2.64) In the analysis, we will require a similar projection operator onto the lowest order Raviart-Thomas velocity space =-=[34, 39]-=-. The RT0 spaces are defined on the unit cube as ẐR(Ê) = ⎛ ⎝ α1 + β1x̂ α2 + β2ŷ α3 + β3ẑ ⎞ ⎠ , ŴR(Ê) = P0(Ê), (2.65) and on the unit square as ẐR(Ê) = ( α1 + β1x̂ α2 + β2ŷ ) , ŴR(Ê) = P0(E... |

190 |
Two families of mixed finite elements for second order elliptic problems.
- Brezzi, Douglas, et al.
- 1985
(Show Context)
Citation Context ...pressure scheme. Similar elimination is achieved in the MFMFE variational framework, by employing appropriate finite element spaces and special quadrature rules. The MFMFE method is based on the BDM1 =-=[9]-=- or the BDDF1 [8] spaces with a trapezoidal quadrature rule applied on the reference element, [21, 44, 47]. We refer to [23, 24] for a related work on quadrilateral grids using a broken Raviart-Thomas... |

101 |
Mixed finite elements for second order elliptic problems in three variables.
- Brezzi, Douglas, et al.
- 1987
(Show Context)
Citation Context ...Similar elimination is achieved in the MFMFE variational framework, by employing appropriate finite element spaces and special quadrature rules. The MFMFE method is based on the BDM1 [9] or the BDDF1 =-=[8]-=- spaces with a trapezoidal quadrature rule applied on the reference element, [21, 44, 47]. We refer to [23, 24] for a related work on quadrilateral grids using a broken Raviart-Thomas space and to [1,... |

83 |
Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods.
- Aavatsmark, Barkve, et al.
- 1998
(Show Context)
Citation Context ... the performance of the MFMFE method for flow on a benchmark test using rough 3D grids and anisotropic coefficients. The MFMFE method was motivated by the multipoint flux approximation (MPFA) methods =-=[2, 3, 13, 14]-=-. In the MPFA finite volume framework, sub-edge (sub-face) fluxes are introduced, which allows for local flux elimination around grid vertices and reduction to a cell-centered pressure scheme. Similar... |

70 | Approximation by quadrilateral finite elements,
- Arnold, Boffi, et al.
- 2002
(Show Context)
Citation Context ...g finite element interpolants or projections. Let Ph be the elliptic elasticity projection in Vh satisfying a(Phu − u, v) = 0, ∀v ∈ Vh. (2.53) The finite element elliptic elasticity theory [12], also =-=[4, 29]-=-, gives ‖u − Phu‖1 h‖u‖2, (2.54) ‖(u − Phu)t‖1 h‖ut‖2. (2.55) It has been shown in [5, 21, 43] that on general quadrilaterals and h2-parallelepipeds, ‖q −sq‖ h‖q‖1. (2.56) However, on general he... |

49 |
An introduction to multipoint flux approximations for quadrilateral grids.
- Aavatsmark
- 2002
(Show Context)
Citation Context ... the performance of the MFMFE method for flow on a benchmark test using rough 3D grids and anisotropic coefficients. The MFMFE method was motivated by the multipoint flux approximation (MPFA) methods =-=[2, 3, 13, 14]-=-. In the MPFA finite volume framework, sub-edge (sub-face) fluxes are introduced, which allows for local flux elimination around grid vertices and reduction to a cell-centered pressure scheme. Similar... |

48 | D.Boffi and R.S.Falk. Quadrilateral H(div) finite elements,
- Arnold
- 2005
(Show Context)
Citation Context ...Vh satisfying a(Phu − u, v) = 0, ∀v ∈ Vh. (2.53) The finite element elliptic elasticity theory [12], also [4, 29], gives ‖u − Phu‖1 h‖u‖2, (2.54) ‖(u − Phu)t‖1 h‖ut‖2. (2.55) It has been shown in =-=[5, 21, 43]-=- that on general quadrilaterals and h2-parallelepipeds, ‖q −sq‖ h‖q‖1. (2.56) However, on general hexahedra, it only holds that [16, 33, 40] ‖q −sq‖ = O(1). (2.57) On simplices, we have optimal inte... |

45 |
Finite Volume Discretization with Imposed Flux Continuity for the General Tensor Pressure Equation,". Computational Geosciences,
- Edwards, Rogers
- 1998
(Show Context)
Citation Context ... the performance of the MFMFE method for flow on a benchmark test using rough 3D grids and anisotropic coefficients. The MFMFE method was motivated by the multipoint flux approximation (MPFA) methods =-=[2, 3, 13, 14]-=-. In the MPFA finite volume framework, sub-edge (sub-face) fluxes are introduced, which allows for local flux elimination around grid vertices and reduction to a cell-centered pressure scheme. Similar... |

22 | Shape functions for velocity interpolation in general hexahedral cells,
- Naff, Russell, et al.
- 2002
(Show Context)
Citation Context ...(2.54) ‖(u − Phu)t‖1 h‖ut‖2. (2.55) It has been shown in [5, 21, 43] that on general quadrilaterals and h2-parallelepipeds, ‖q −sq‖ h‖q‖1. (2.56) However, on general hexahedra, it only holds that =-=[16, 33, 40]-=- ‖q −sq‖ = O(1). (2.57) On simplices, we have optimal interpolation error estimates [9]: ‖q −sq‖ hr‖q‖r , r = 1, 2. (2.58) Using a scaling argument and the Bramble-Hilbert lemma [12], it can be show... |

19 |
Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids.
- Edwards
- 2002
(Show Context)
Citation Context |

19 | Convergence of multi point flux approximations on quadrilateral grids. Numerical Methods for Partial Differential Equations,
- Klausen, Winther
- 2006
(Show Context)
Citation Context ...t spaces and special quadrature rules. The MFMFE method is based on the BDM1 [9] or the BDDF1 [8] spaces with a trapezoidal quadrature rule applied on the reference element, [21, 44, 47]. We refer to =-=[23, 24]-=- for a related work on quadrilateral grids using a broken Raviart-Thomas space and to [1, 22] for papers utilizing both approaches. The choice of CG for elasticity is reasonable if the permeability is... |

17 | Convergence of a symmetric MPFA method on quadrilateral grids
- Aavvatsmark, Eigestad, et al.
- 2005
(Show Context)
Citation Context ...[8] spaces with a trapezoidal quadrature rule applied on the reference element, [21, 44, 47]. We refer to [23, 24] for a related work on quadrilateral grids using a broken Raviart-Thomas space and to =-=[1, 22]-=- for papers utilizing both approaches. The choice of CG for elasticity is reasonable if the permeability is not very small and locking is not an issue. As mentioned above, there has been previous work... |

17 |
Robust convergence of multipoint flux approximation on rough grids. Numerische Mathematik 2006
- RA, Winther
(Show Context)
Citation Context ...t spaces and special quadrature rules. The MFMFE method is based on the BDM1 [9] or the BDDF1 [8] spaces with a trapezoidal quadrature rule applied on the reference element, [21, 44, 47]. We refer to =-=[23, 24]-=- for a related work on quadrilateral grids using a broken Raviart-Thomas space and to [1, 22] for papers utilizing both approaches. The choice of CG for elasticity is reasonable if the permeability is... |

13 | On stability and convergence of finite element approximations of Biot's consolidation problem - Murad, Loula - 1994 |

12 | G.: A least-squares mixed finite element method for Biot’s consolidation problem in porous media
- Korsawe, Starke
- 2005
(Show Context)
Citation Context ...lor-Hood finite elements are employed for a displacement–pressure variational formulation. A least squares formulation that approximates directly the solid stress and the fluid velocity is studied in =-=[25, 26]-=-. Finite difference schemes on staggered grids designed to avoid nonphysical oscillations at early times have been developed in 1D in [15, 19]. The method in [15] can handle discontinuous coefficients... |

11 |
A coupled geomechanics and reservoir flow model on parallel computers
- Gai
- 2003
(Show Context)
Citation Context ...stems are coupled using DG jumps and mortars. 58 Comput Geosci (2014) 18:57–75 Applications of the Biot system to the computational modeling of coupled reservoir flow and geomechanics can be found in =-=[11, 17, 18, 41]-=-. The focus of this paper is to develop a discretization method for the poroelasticity system that is suitable for irregular and rough grids and discontinuous full tensor permeabilities that are often... |

11 |
A finite difference analysis of Biot’s consolidation model
- Gaspar, Lisbona, et al.
(Show Context)
Citation Context ...tly the solid stress and the fluid velocity is studied in [25, 26]. Finite difference schemes on staggered grids designed to avoid nonphysical oscillations at early times have been developed in 1D in =-=[15, 19]-=-. The method in [15] can handle discontinuous coefficients through harmonic averaging. A formulation based on mixed finite element (MFE) methods for flow and continuous Galerkin (CG) for elasticity ha... |

11 |
The discontinuous Galerkin finite element method for the 2d shallow water equations. Mathematics and Computers in Simulation
- Li, RX
- 2001
(Show Context)
Citation Context ...luid velocity is computed directly. Further work addresses the problem of eliminating locking or removal of nonphysical pressure oscillations via the use of discontinuous Galerkin (DG) for elasticity =-=[27, 28, 38]-=-. In [20], a parallel domain decomposition method has been developed for coupling a time-dependent poroelastic model in a localized region with an elastic model in adjacent regions. Each model is disc... |

10 | A multipoint flux mixed finite element method on hexahedra
- Ingram, Wheeler, et al.
- 2010
(Show Context)
Citation Context ...urface flows. To this end, we develop a formulation that couples multipoint flux mixed finite element (MFMFE) methods for flow with CG for elasticity. The MFMFE method was developed for Darcy flow in =-=[21, 44, 47]-=-. It is locally conservative with continuous fluxes and can be viewed within a variational framework as a mixed finite element method with special approximating spaces and quadrature rules. The MFMFE ... |

10 | Improved accuracy in finite element analysis of Biot's consolidation problem - Murad, Loula - 1992 |

9 |
Finite element analysis of poroelastic consolidation in porous media: mixed and standard approaches. University of Hannover–Center for Applied Geosciences
- KORSAWE, STARKE, et al.
- 2003
(Show Context)
Citation Context ...lor-Hood finite elements are employed for a displacement–pressure variational formulation. A least squares formulation that approximates directly the solid stress and the fluid velocity is studied in =-=[25, 26]-=-. Finite difference schemes on staggered grids designed to avoid nonphysical oscillations at early times have been developed in 1D in [15, 19]. The method in [15] can handle discontinuous coefficients... |

8 | Mapped finite elements on hexahedra. Necessary and sufficient conditions for optimal interpolation errors,
- Matthies
- 2001
(Show Context)
Citation Context ...g finite element interpolants or projections. Let Ph be the elliptic elasticity projection in Vh satisfying a(Phu − u, v) = 0, ∀v ∈ Vh. (2.53) The finite element elliptic elasticity theory [12], also =-=[4, 29]-=-, gives ‖u − Phu‖1 h‖u‖2, (2.54) ‖(u − Phu)t‖1 h‖ut‖2. (2.55) It has been shown in [5, 21, 43] that on general quadrilaterals and h2-parallelepipeds, ‖q −sq‖ h‖q‖1. (2.56) However, on general he... |

8 | Asymptotic behavior of semidiscrete finite-element approximations of Biot’s consolidation problem - Murad, Thomée, et al. - 1996 |

7 |
Coupled geomechanical and reservoir modeling on parallel computers
- Gai, Dean, et al.
- 2003
(Show Context)
Citation Context ...stems are coupled using DG jumps and mortars. 58 Comput Geosci (2014) 18:57–75 Applications of the Biot system to the computational modeling of coupled reservoir flow and geomechanics can be found in =-=[11, 17, 18, 41]-=-. The focus of this paper is to develop a discretization method for the poroelasticity system that is suitable for irregular and rough grids and discontinuous full tensor permeabilities that are often... |

6 |
A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: the discrete in time case
- Phillips, Wheeler
- 2007
(Show Context)
Citation Context ...] can handle discontinuous coefficients through harmonic averaging. A formulation based on mixed finite element (MFE) methods for flow and continuous Galerkin (CG) for elasticity has been proposed in =-=[36, 37]-=-. The advantage of this approach is that the fluid approximation is locally mass conservative and the fluid velocity is computed directly. Further work addresses the problem of eliminating locking or ... |

5 | Hexahedral H(div) and H(curl) finite elements
- Falk, Gatto, et al.
(Show Context)
Citation Context ...(2.54) ‖(u − Phu)t‖1 h‖ut‖2. (2.55) It has been shown in [5, 21, 43] that on general quadrilaterals and h2-parallelepipeds, ‖q −sq‖ h‖q‖1. (2.56) However, on general hexahedra, it only holds that =-=[16, 33, 40]-=- ‖q −sq‖ = O(1). (2.57) On simplices, we have optimal interpolation error estimates [9]: ‖q −sq‖ hr‖q‖r , r = 1, 2. (2.58) Using a scaling argument and the Bramble-Hilbert lemma [12], it can be show... |

4 |
Domain decomposition for poroelasticity and elasticity with DG jumps and mortars
- Girault, Pencheva, et al.
(Show Context)
Citation Context ...computed directly. Further work addresses the problem of eliminating locking or removal of nonphysical pressure oscillations via the use of discontinuous Galerkin (DG) for elasticity [27, 28, 38]. In =-=[20]-=-, a parallel domain decomposition method has been developed for coupling a time-dependent poroelastic model in a localized region with an elastic model in adjacent regions. Each model is discretized i... |

4 |
A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity,"
- Phillips, Wheeler
- 2008
(Show Context)
Citation Context ...luid velocity is computed directly. Further work addresses the problem of eliminating locking or removal of nonphysical pressure oscillations via the use of discontinuous Galerkin (DG) for elasticity =-=[27, 28, 38]-=-. In [20], a parallel domain decomposition method has been developed for coupling a time-dependent poroelastic model in a localized region with an elastic model in adjacent regions. Each model is disc... |

3 |
Iterative coupled analysis of geomechanics and fluid flow for rock compaction in reservoir simulation
- Chin, Thomas, et al.
- 2002
(Show Context)
Citation Context ...stems are coupled using DG jumps and mortars. 58 Comput Geosci (2014) 18:57–75 Applications of the Biot system to the computational modeling of coupled reservoir flow and geomechanics can be found in =-=[11, 17, 18, 41]-=-. The focus of this paper is to develop a discretization method for the poroelasticity system that is suitable for irregular and rough grids and discontinuous full tensor permeabilities that are often... |

3 |
Finite element methods in linear poroelasticity: theoretical and computational results
- Phillips
- 2005
(Show Context)
Citation Context ...th curved boundaries using both quadrilateral and triangular meshes as shown in Fig. 3. The elasticity boundary conditions are motivated by the cantilever bracket problem. This problem was studied in =-=[27, 35]-=- for a fluid saturated bracket using the system of poroelasticity (2.1)–(2.3). The elasticity boundary conditions are u = 0 on 1, σn = 0 on 2 ∪ 3, σn = (0,−1)T on 4, where 1 and 2 are the left a... |

2 |
A.: On convergence of certain finite volume difference discretizations for 1D poroelasticity interface problems
- Ewing, Iliev, et al.
- 2007
(Show Context)
Citation Context ...tly the solid stress and the fluid velocity is studied in [25, 26]. Finite difference schemes on staggered grids designed to avoid nonphysical oscillations at early times have been developed in 1D in =-=[15, 19]-=-. The method in [15] can handle discontinuous coefficients through harmonic averaging. A formulation based on mixed finite element (MFE) methods for flow and continuous Galerkin (CG) for elasticity ha... |

2 |
On a coupled discontinuous/continuous Galerkin framework and an adaptive penalty scheme for poroelasticity problems, Computer Methods in Applied Mechanics and Engineering
- Liu, Wheeler, et al.
- 2009
(Show Context)
Citation Context ...luid velocity is computed directly. Further work addresses the problem of eliminating locking or removal of nonphysical pressure oscillations via the use of discontinuous Galerkin (DG) for elasticity =-=[27, 28, 38]-=-. In [20], a parallel domain decomposition method has been developed for coupling a time-dependent poroelastic model in a localized region with an elastic model in adjacent regions. Each model is disc... |

1 | Eigestad, G.T.: Convergence of MPFA on triangulations and for Richards’ equation - Klausen, Radu - 2008 |