A new viewpoint on mixed elements

Bossavit, Alain

doi:10.1007/bf00452998

Cited by 16 publications

(8 citation statements)

References 17 publications

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“…Fix an integer k ≥ 0. We then recall the following spaces (the spaces Q h and V h were introduced in [15]; see also [3] and [14] for the connection between these spaces and differential forms, which illuminates many of their properties):…”

Section: Finite Element Discretizationmentioning

confidence: 99%

Multigrid in H (div) and H (curl)

2000

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We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the Hilbert spaces H(div) and H(curl ) in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V-cycle is an efficient solver and preconditioner for the discrete operator. All results are uniform with respect to the mesh size, the number of mesh levels, and weights on the two terms in the inner products. Subject Classification (1991): 65N55, 65N30 Mathematics

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Section: Finite Element Discretizationmentioning

confidence: 99%

Multigrid in H (div) and H (curl)

2000

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“…(ii) We discretize the skeleton displacement field ( u ) using standard node‐based finite elements in such a way that together with the fluid pressure field, the LBB condition in the undrained limit is satisfied in the same manner as in incompressible elasticity problems . In the context of RT elements, we note other topology‐inspired finite elements …”

Section: Introductionmentioning

confidence: 99%

“…29,49 In the context of RT elements, we note other topology-inspired finite elements. 45,50,51 Most finite elements of the three-field category for porous media are obtained using standard nodal discretization of all three fields. 2,14,27,[52][53][54][55][56][57][58] However, these finite elements usually fail to fulfill the LBB condition.…”

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confidence: 99%

Mixed finite element formulation for dynamics of porous media

Lotfian

Sivaselvan

2018

Numerical Meth Engineering

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Summary This paper presents a numerical approach for computing solutions to Biot's fully dynamic model of saturated porous media with incompressible solid and fluid phases. Spatial discretization is based on a three‐field (u‐w‐p) formulation, employing a lowest‐order Raviart‐Thomas mixed element for the fluid Darcy velocity (w) and pressure (p) fields, and a nodal finite element for skeleton displacement field (u). The discretization is constructed based on the natural topology of the variables and satisfies the Ladyženskaja‐Babuška‐Brezzi stability condition, avoiding locking in the incompressible and undrained limits. Since fluid acceleration is not neglected, the three‐field formulation fully captures dynamic behavior even under high‐frequency loading phenomena. The importance of consistent initial conditions is addressed for poroelasticity equations with the mass balance constraint, a system of differential algebraic equations. Energy balance is derived for the porous medium and used to assess accuracy of time integration. To demonstrate the performance of the proposed approach, a variety of numerical studies are carried out including verification with analytical and boundary element solutions and analyses of wave propagation, effect of hydraulic conductivity on damping and frequency content, energy balance, mass lumping, mesh pattern and size, and stability. Some discrepancies found in dynamic poroelasticity results in the literature are also explained.

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“…At first glance they have little in common, but a closer scrutiny reveals that they are all specimens of discrete differential forms [21,4]. As such they are members of a family of finite element schemes that also includes the conventional Lagrangian H 1 (Ω)-conforming elements (as discrete 0-forms).…”

Section: Introductionmentioning

confidence: 98%

Mixed finite elements on sparse grids

Gradinaru

Hiptmair

2003

Numerische Mathematik

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This paper generalizes the idea of approximation on sparse grids to discrete differential forms that include H(div; Ω)-and H(curl; Ω)-conforming mixed finite element spaces as special cases. We elaborate on the construction of the spaces, introduce suitable nodal interpolation operators on sparse grids and establish their approximation properties. We discuss how nodal interpolation operators can be approximated. The stability of H(div; Ω)-conforming finite elements on sparse grids, when used to approximate second order elliptic problems in mixed formulation, is investigated both theoretically and in numerical experiments. (2000): 65N30 41A10 58A15 Mathematics Subject Classification

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A new viewpoint on mixed elements

Cited by 16 publications

References 17 publications

Multigrid in H (div) and H (curl)

Multigrid in H (div) and H (curl)

Mixed finite element formulation for dynamics of porous media

Mixed finite elements on sparse grids

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