A Posteriori Error Estimator for Obstacle Problems

Weiss, André; Wohlmuth, Barbara

doi:10.1137/090773921

Cited by 22 publications

(13 citation statements)

References 36 publications

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“…Using such a post-processed Lagrange multiplier on the right side of the vertex-based equation system for the flux moments gives much better results. Details can be found in Weiss and Wohlmuth (2010). Following the construction principle of Section 6 and applying mixed RT 0 , RT 1 or BDM 1 elements will result in robust and reliable adaptive mesh refinement in the case of smooth obstacles.…”

Section: Mathematical Finance: American Optionsmentioning

confidence: 99%

“…In addition, we apply the previously discussed modification in the error indicator, because of the kinks in the pay-off function, and take note of the different structure of the PDE (8.2) compared to the Laplace operator. Two different pay-off functions are tested, ψ max := max(0, K − max(x 1 , x 2 )) and ψ min := max(0, K − min(x 1 , x 2 )), and we refer to Weiss and Wohlmuth (2010) for the problem specification. The adaptively refined meshes in Figure 8.3 show that the error estimator does not over-refine at the kinks of the pay-off functions and that the proposed modification also works well for much more complex situations.…”

Section: American Basket Optionmentioning

confidence: 99%

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Variationally consistent discretization schemes and numerical algorithms for contact problems

Wohlmuth¹

2011

Acta Numerica

199

187

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We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities, such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance. The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain. Having a uniform inf-sup bound, one can then establish optimal low-ordera prioriconvergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations. Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal–dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual- and equilibrated-baseda posteriorierror estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time-integration schemes. However, the differential algebraic character of the system can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.

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Section: Mathematical Finance: American Optionsmentioning

confidence: 99%

Section: American Basket Optionmentioning

confidence: 99%

Variationally consistent discretization schemes and numerical algorithms for contact problems

Wohlmuth¹

2011

Acta Numerica

199

187

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“…Moreover, we will sketch how stress-splitting approaches [20] can be incorporated into the error estimates. Various methods for a posteriori error estimation of the obstacle problem can be found in the literature, see, e.g., [4,10,15,25,32,35]. Here we focus on the approach by [25] utilizing a suitable Galerkin functional and a useful definition of the discrete constraining forces.…”

Section: Introductionmentioning

confidence: 99%

A posteriori estimator for the adaptive solution of a quasi-static fracture phase-field model with irreversibility constraints

Walloth,

Wollner

2021

Preprint

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Within this article, we develop a residual type a posteriori error estimator for a time discrete quasi-static phase-field fracture model. Particular emphasize is given to the robustness of the error estimator for the variational inequality governing the phase-field evolution with respect to the phase-field regularization parameter . The article concludes with numerical examples highlighting the performance of the proposed a posteriori error estimators on three standard test cases; the single edge notched tension and shear test as well as the L-shaped panel test.

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“…An intense activity is related to multiscale and mortar elements [PVWW13,TW13] as well as to porous media and porous elasticity [MN17, RDPE + 17, VY18]. Obstacle and contact problems have been studied in [BHS08,WW10,HW12]. Further examples of applications include Maxwell's equations [CNT17], hp finite elements [DEV16], and eigenvalue problems [CDM + 17, LO13, BBS19].…”

Section: Introductionmentioning

confidence: 99%

The Prager–Synge theorem in reconstruction based a posteriori error estimation

Bertrand¹,

Boffi²

2020

75 Years of Mathematics Of Computation

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In this paper we review the hypercircle method of Prager and Synge. This theory inspired several studies and induced an active research in the area of a posteriori error analysis. In particular, we review the Braess-Schöberl error estimator in the context of the Poisson problem. We discuss adaptive finite element schemes based on two variants of the estimator and we prove the convergence and optimality of the resulting algorithms.

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A Posteriori Error Estimator for Obstacle Problems

Cited by 22 publications

References 36 publications

Variationally consistent discretization schemes and numerical algorithms for contact problems

Variationally consistent discretization schemes and numerical algorithms for contact problems

A posteriori estimator for the adaptive solution of a quasi-static fracture phase-field model with irreversibility constraints

The Prager–Synge theorem in reconstruction based a posteriori error estimation

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