An a posteriori error estimate for a linear-nonlinear transmission problem in plane elastostatics

Barrientos, Mauricio A.; Gatica, Gabriel N.; Heuer, Norbert

doi:10.1007/s100920200005

Cited by 6 publications

(8 citation statements)

References 26 publications

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“…Since it is a linear problem, we suggest to apply the p or the h − p version, as indicated in [27]. Alternatively, we propose in the next subsection a fully explicit reliable a posteriori error estimate, which does not require the approximate solution of the local problems, and which is based on an appropriate choice of the functionφ ϕ ϕ h (see also [10]). The bound given by (5.25) plays here an essential role.…”

Section: The Main a Posteriori Error Estimatementioning

confidence: 98%

“…Here we follow the analysis from [9] (see also [10], [11], and [26]) and derive a new mixed variational formulation for problem (2.3). Let us first define the tensor space…”

Section: A a Twofold Saddle Point Operator Equationmentioning

confidence: 99%

“…We proved in [11] that one can combine the classical Bank-Weiser method from [15] with the ideas from [5] and [7], to derive an a posteriori error estimate for the twofold saddle point formulation of an hyperelastic material in the plane. This analysis was recently extended in [16] and [10] to linear and nonlinear transmission problems in plane elastostatics. We prove in this article that it can also be successfully applied to nonlinear incompressible materials.…”

Section: Introductionmentioning

confidence: 96%

“…More precisely, we consider the same non-Newtonian flow as in [7] and [8] and apply the dual-mixed approach from [9] (see also [10] and [11]), to study the solvability and finite element approximations of the corresponding variational formulation. This approach, also called expanded mixed finite element method in the literature, is based on the introduction of the strain tensor as an additional unknown, which yields a two fold saddle point operator equation as the associated weak formulation.…”

Section: Introductionmentioning

confidence: 99%

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A mixed‐FEM formulation for nonlinear incompressible elasticity in the plane

Gatica

Stephan

2001

Numerical Methods Partial

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This article deals with an expanded mixed finite element formulation, based on the Hu-Washizu principle, for a nonlinear incompressible material in the plane. We follow our related previous works and introduce both the stress and the strain tensors as further unknowns, which yields a two-fold saddle point operator equation as the corresponding variational formulation. A slight generalization of the classical Babuška-Brezzi's theory is applied to prove unique solvability of the continuous and discrete formulations, and to derive the corresponding a priori error analysis. An extension of the well-known PEERS space is used to define an stable associated Galerkin scheme. Finally, we provide an a posteriori error analysis based on the classical Bank-Weiser approach.

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Section: The Main a Posteriori Error Estimatementioning

confidence: 98%

“…Here we follow the analysis from [9] (see also [10], [11], and [26]) and derive a new mixed variational formulation for problem (2.3). Let us first define the tensor space…”

Section: A a Twofold Saddle Point Operator Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A mixed‐FEM formulation for nonlinear incompressible elasticity in the plane

Gatica

Stephan

2001

Numerical Methods Partial

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“…To this respect, concerning the combination of the usual FEM with BEM, we may refer to [10,13,14], where mainly reliable a-posteriori error estimates are provided. More recently, this kind of result has been extended to the coupling of dual-mixed FEM and BEM for linear and nonlinear problems (see [5,6,12,19,21,23]). Here, the estimates for the linear problems are of explicit residual type, and those for the nonlinear ones are based on the classical Bank-Weiser implicit approach.…”

mentioning

confidence: 99%

A posteriorierror estimates for linear exterior problemsviamixed-FEM and DtN mappings

2002

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Abstract. In this paper we combine the dual-mixed finite element method with a Dirichlet-toNeumann mapping (given in terms of a boundary integral operator) to solve linear exterior transmission problems in the plane. As a model we consider a second order elliptic equation in divergence form coupled with the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive the usual Cea error estimate and the corresponding rate of convergence. In addition, we develop two different a-posteriori error analyses yielding explicit residual and implicit Bank-Weiser type reliable estimates, respectively. Several numerical results illustrate the suitability of these estimators for the adaptive computation of the discrete solutions.Mathematics Subject Classification. 65N15, 65N30, 65N38, 65N50.

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A posteriori error estimates for the mixed finite element method with Lagrange multipliers

Gatica

Maischak

2004

Numerical Methods Partial

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We consider the mixed finite element method with Lagrange multipliers as applied to second-order elliptic equations in divergence form with mixed boundary conditions. The corresponding Galerkin scheme is defined by using Raviart-Thomas spaces. We develop a posteriori error analyses yielding a reliable and efficient estimate based on residuals, and a reliable and quasi-efficient estimate based on local problems, respectively. Several numerical results illustrate the suitability of these a posteriori estimates for the adaptive computation of the discrete solutions.

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An a posteriori error estimate for a linear-nonlinear transmission problem in plane elastostatics

Cited by 6 publications

References 26 publications

A mixed‐FEM formulation for nonlinear incompressible elasticity in the plane

A mixed‐FEM formulation for nonlinear incompressible elasticity in the plane

A posteriorierror estimates for linear exterior problemsviamixed-FEM and DtN mappings

A posteriori error estimates for the mixed finite element method with Lagrange multipliers

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