Error Analysis of Galerkin Least Squares Methods for the Elasticity Equations

Franca, Leopoldo P.; Stenberg, Rolf

doi:10.1137/0728084

Cited by 243 publications

(184 citation statements)

References 23 publications

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“…Again, it follows that there is no further restriction on κ 2 besides being positive. Next, Korn's first inequality establishes now the existence of k D ∈ (0, 1), depending only on Ω and 22) and hence (3.21) yields…”

Section: Mixed Boundary Conditionsmentioning

confidence: 99%

“…They are also known as Galerkin least-squares methods and have already been extended in different directions. Some applications to elasticity problems can be found in [14,22], and a non-symmetric variant was considered in [17] for the Stokes problem. In addition, stabilized mixed finite element methods for related problems, including Darcy flow, incompressible flows, plates, and shells, can be seen in [2,9,15,18,21,26,27,29,31].…”

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confidence: 99%

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Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

Gatica¹

2006

ESAIM: M2AN

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Abstract. We present a new stabilized mixed finite element method for the linear elasticity problem in R 2 . The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive and equilibrium equations, and from the relation defining the rotation in terms of the displacement. We show that the resulting augmented variational formulation and the associated Galerkin scheme are well posed, and that the latter becomes locking-free and asymptotically locking-free for Dirichlet and mixed boundary conditions, respectively. In particular, the discrete scheme allows the utilization of Raviart-Thomas spaces of lowest order for the stress tensor, piecewise linear elements for the displacement, and piecewise constants for the rotation. In the case of mixed boundary conditions, the essential one (Neumann) is imposed weakly, which yields the introduction of the trace of the displacement as a suitable Lagrange multiplier. This trace is then approximated by piecewise linear elements on an independent partition of the Neumann boundary whose mesh size needs to satisfy a compatibility condition with the mesh size associated to the triangulation of the domain. Several numerical results illustrating the good performance of the augmented mixed finite element scheme in the case of Dirichlet boundary conditions are also reported.Mathematics Subject Classification. 65N12, 65N15, 65N30, 74B05.

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Section: Mixed Boundary Conditionsmentioning

confidence: 99%

mentioning

confidence: 99%

Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

Gatica¹

2006

ESAIM: M2AN

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“…We first begin by a (partial) stability result for the discrete gradient operator, which is not specific to a clustered approximation, and may be seen as the "finite volume analogue" to a lemma already known in the finite element context [17], Lemma 3.3. …”

Section: Stability Of the Schemementioning

confidence: 99%

Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows

Eymard¹,

Herbin²,

Latché³

et al. 2009

ESAIM: M2AN

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Abstract. We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e., in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem. An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh. Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms. Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L 2 norm.Mathematics Subject Classification. 65N12, 65N15, 65N30, 76D05, 76D07, 76M25.

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“…Mixed finite element schemes for this version have been recently proposed by Fortin and Pierre [1989] and Franca and Stenberg [1991] on the Maxwell model corresponding to a = 1 and by Baranger and Sandri [1992] for a e [0,1). For the approximation accuracy, only have the usual estimâtes been obtained in literature.…”

Section: Introductionmentioning

confidence: 99%

Global superconvergence approximations of the mixed finite element method for the Stokes problem and the linear elasticity equation

Zhou¹

1996

ESAIM: M2AN

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Error Analysis of Galerkin Least Squares Methods for the Elasticity Equations

Cited by 243 publications

References 23 publications

Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows

Global superconvergence approximations of the mixed finite element method for the Stokes problem and the linear elasticity equation

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