A posteriori error estimation and adaptive solution of elliptic variational inequalities of the second kind

We propose and analyze the finite volume method for solving the variational inequalities of first and second kinds. The stability and convergence analysis are given for this method. For the elliptic obstacle problem, we derive the optimal error estimate in the H1‐norm. For the simplified friction problem, we establish an abstract H1‐error estimate, which implies the convergence if the exact solution u∈H1(Ω) and the optimal error estimate if u∈H1 + α(Ω),0 < α≤2. Copyright © 2015 John Wiley & Sons, Ltd.

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“…Table gives the numerical results with successively halved mesh size h . We see that a convergence rate of

O (h^{\frac{1}{2}})

‐order is achieved as the theoretical prediction and this convergence rate is basically consistent with that of the FEM .…”

Section: Numerical Examplesupporting

confidence: 83%

Finite volume method for the variational inequalities of first and second kinds

Zhang

Tang

2015

Math Methods in App Sciences

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“…Adaptive techniques based on a posteriori error estimators have become an indispensable tool and are well established for such methods; see [2,27,55] and the references therein. Error estimators have also been successfully used in the field of nonconforming discretizations [1,19,20,24] and for variational inequalities [3,13,26] such as obstacle problems; see [14,32,34,53] and for contact problems, see [11,12,18,22,31,41,61].…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimator and error control for contact problems

Weiss

Wohlmuth

2009

Math. Comp.

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Abstract. In this paper, we consider two error estimators for one-body contact problems. The first error estimator is defined in terms of H(div)-conforming stress approximations and equilibrated fluxes while the second is a standard edge-based residual error estimator without any modification with respect to the contact. We show reliability and efficiency for both estimators. Moreover, the error is bounded by the first estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order. The second estimator is used in a control-based AFEM refinement strategy, and the decay of the error in the energy is shown. Several numerical tests demonstrate the performance of both estimators.

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“…To develop adaptive solution algorithms, it is necessary to first derive a posteriori error estimate for the numerical method (cf. [30][31][32][33][34][35][36] on a posteriori error analysis and adaptive solution algorithms for simpler models of elliptic or parabolic variational inequalities). Success of the work on these two aspects will not only lead to more efficient and effective numerical methods for solving the contact problem of this paper, but also be useful in numerically solving other history-dependent contact problems.…”

Section: Discussionmentioning

confidence: 99%