A posteriori error estimates for a mixed-FEM formulation of a non-linear elliptic problem

Araya, Rodolfo; Barrios, Tomás P.; Gatica, Gabriel N.; Heuer, Norbert

doi:10.1016/s0045-7825(01)00414-5

Cited by 12 publications

(14 citation statements)

References 21 publications

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“…These estimations are the essential component in the design of a reliable and efficient algorithm, as we can notice in the numerical examples. In future researchs we will try to extend this study to non-linear problems combining the duality ( [12,13,9,21]) with the advantages provided by the mixed methods for dealing with this kind of problems ( [2,5,16,18]). …”

Section: Discussionmentioning

confidence: 99%

Space-Time adaptive algorithm for the mixed parabolic problem

2006

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In this paper we present an a-posteriori error estimator for the mixed formulation of a linear parabolic problem, used for designing an efficient adaptive algorithm. Our space-time discretization consists of lowest order Raviart-Thomas finite element over graded meshes and discontinuous Galerkin method with variable time step. Finally, several examples show that the proposed method is efficient and reliable.

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Section: Discussionmentioning

confidence: 99%

Space-Time adaptive algorithm for the mixed parabolic problem

2006

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“…We mention as representative the work of Ainsworth and Oden (1997) where global error estimators for the Stokes and Oseen equations are derived. Another global residual a posteriori error estimator for the first Piola-Kirchhoff stress error in (nearly) incompressible finite hyperelasticity was introduced by Brink and Stein (1998); see also Araya et al (2002) for further global error estimators for nonlinear elliptic problems. Moreover, a goal-oriented a posteriori error estimator for the Stokes and Oseen equations, which rests upon a Ritz projection similar to (274) was presented by Oden and Prudhomme (1999).…”

Section: Finite Element Methods For Elasticity With Error-controlled mentioning

confidence: 99%

Finite Element Methods for Elasticity with Error‐Controlled Discretization and Model Adaptivity

Stein¹,

Rüter²

2004

Encyclopedia of Computational Mechanics

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The essential topics of the finite element method for linear and finite elastic deformations of solids are presented in this chapter from both the mechanical and mathematical point of view. As a starting point, the nonlinear, and linearized theory of elasticity are derived in a rigorous way, followed by the classical variational principles of elasticity, which are the basis for the finite element method in its various forms. More precisely, the discrete variational approach of the (one‐field) Dirichlet minimization principle of the total potential energy is presented, followed by concise representations of the (two‐field) Hellinger–Reissner stationary dual‐mixed principle and the (three‐field) Hu–Washizu stationary mixed principle, including the main features of the associated finite element methods. The main objective of this chapter is the systematic treatment of error estimation procedures and adaptivity for the linearized and finite elasticity problem covering both global and goal‐oriented a posteriori error estimators. The three basic classes, that is, residual‐, hierarchical‐, and averaging‐type error estimators are presented and applied to a fracture mechanics problem as an example. A further challenging topic is the combination of error‐controlled adaptive finite element solutions with hierarchical model and dimension adaptivity of the underlying mathematical model, especially for thin‐walled structures where model expansion is necessary in subdomains with boundary layers and other disturbances.

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“…− Ω ζ 0 div σ dx + Ω a 0 (·, θ)ζ 0 dx = Ω f ζ 0 dx, (2.2) for all ζ := (ζ 0 , ζ 1 , ζ 2 ) T ∈ X 1 := [L 2 (Ω)] 3 . Also, we multiply the relation (θ 1 , θ 2 ) T = ∇u = ∇θ 0 by a test function τ ∈ M 1 := H (div; Ω), integrate by parts on Ω, and use that u = 0 on Γ D , to obtain − Ω θ 0 div τ dx − Ω (θ 1 , θ 2 ) T · τ dx + τ · ν, ξ Γ N = 0 ∀τ ∈ M 1 , (2.3) where ξ := u| Γ N ∈ M := H 1/2 00 (Γ N ) is a further unknown acting as a Lagrange multiplier.…”

Section: The Mixed Variational Formulationmentioning

confidence: 99%