Analysis of a Class of Penalty Methods for Computing Singular Minimizers

Carstensen, Carsten; Ortner, Christoph

doi:10.2478/cmam-2010-0008

Cited by 11 publications

(12 citation statements)

References 26 publications

(38 reference statements)

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“…In this sense, we remark that in order to prove the convergence of the finite-element method, it is necessary to have the density of the Lipschitz functions in the space where we are looking for the solution of the state equation. A reciprocate of this result has been obtained in [5] for a calculus of variations problem without restrictions.…”

mentioning

confidence: 82%

Numerical Approximation of a One-Dimensional Elliptic Optimal Design Problem

Casado–Díaz¹,

Castro²,

Luna-Laynez³

et al. 2011

Multiscale Model. Simul.

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Abstract. We address the numerical approximation by finite-element methods of an optimal design problem for a two phase material in one space dimension. This problem, in the continuous setting, due to high frequency oscillations, often does not have a classical solution, and a relaxed formulation is needed to ensure existence. On the contrary, the discrete versions obtained by numerical approximation have a solution. In this article we prove the convergence of the discretizations and obtain convergence rates. We also show a faster convergence when the relaxed version of the continuous problem is taken into account when building the discretization strategy. In particular it is worth emphasizing that, even when the original problem has a classical solution so that relaxation is not necessary, numerical algorithms converge faster when implemented on the relaxed version.

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confidence: 82%

Numerical Approximation of a One-Dimensional Elliptic Optimal Design Problem

Casado–Díaz¹,

Castro²,

Luna-Laynez³

et al. 2011

Multiscale Model. Simul.

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“…In particular, one cannot hope for convergence of conformal polynomial finite elements as far as discretization of y concerns, which are always only subspaces of W 1,∞ (Ω; R d ), which was already well recognized for static problems e.g. in [BaL06,CaO10,Li95,Li96]. For quasistatic rate-independent problems as considered here, this difficulty is still present.…”

Section: Numerical Approximationmentioning

confidence: 92%

“…Thus we make a penalization/regularization by a parameter ε > 0 of the stored energy similarly as in [CaO10] and also of the global constraints by considering…”

Section: Numerical Approximationmentioning

confidence: 99%

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Rate-independent elastoplasticity at finite strains and its numerical approximation

Mielke

Roubíček

2016

Math. Models Methods Appl. Sci.

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Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation is approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The nonself-penetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously bypasses the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme toward energetic solutions. In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations.

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“…Moreover, a more striking property, which was proved by Ball and Knowles (cf. [2]), is that if u j is a sequence of functions in W 1,q (0, 1) for q ≥ 3 2 with u j (0) = 0 and u j (1) = 1 such that u j → u a.e. as j → ∞, then J (u j ) → ∞ as j → ∞.…”

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confidence: 99%

An Enhanced Finite Element Method for a Class of Variational Problems Exhibiting the Lavrentiev Gap Phenomenon

Feng¹,

Schnake²

2018

CiCP

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This paper develops an enhanced finite element method for approximating a class of variational problems which exhibit the Lavrentiev gap phenomenon in the sense that the minimum values of the energy functional have a nontrivial gap when the functional is minimized on spaces W 1,1 and W 1,∞ . To remedy the standard finite element method, which fails to converge for such variational problems, a simple and effective cut-off procedure is utilized to design the (enhanced finite element) discrete energy functional. In essence the proposed discrete energy functional curbs the gap phenomenon by capping the derivatives of its input on a scale of O(h −α ) (where h denotes the mesh size) for some positive constant α. A sufficient condition is proposed for determining the problemdependent parameter α. Extensive 1-D and 2-D numerical experiment results are provided to show the convergence behavior and the performance of the proposed enhanced finite element method.

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Analysis of a Class of Penalty Methods for Computing Singular Minimizers

Cited by 11 publications

References 26 publications

Numerical Approximation of a One-Dimensional Elliptic Optimal Design Problem

Numerical Approximation of a One-Dimensional Elliptic Optimal Design Problem

Rate-independent elastoplasticity at finite strains and its numerical approximation

An Enhanced Finite Element Method for a Class of Variational Problems Exhibiting the Lavrentiev Gap Phenomenon

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