Numerical solutions of a two-phase membrane problem

Bozorgnia, Farid

doi:10.1016/j.apnum.2010.08.007

Cited by 13 publications

(22 citation statements)

References 15 publications

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“…This subsection extends results of [7]. We consider the two-phase obstacle problem in 1D from [1]. Here, Ω = (−1, 1), f = 0, A = I, α ⊕ = α = 8 and the Dirichlet boundary conditions u(−1) = −1, u(1) = 1.…”

Section: The Two-phase Obstacle Problem In 1dmentioning

confidence: 65%

Verifications of Primal Energy Identities for Variational Problems with Obstacles

Repin

Valdman²

2018

Large-Scale Scientific Computing

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We discuss error identities for two classes of free boundary problems generated by obstacles. The identities suggest true forms of the respective error measures which consist of two parts: standard energy norm and a certain nonlinear measure. The latter measure controls (in a weak sense) approximation of free boundaries. Numerical tests confirm sharpness of error identities and show that in different examples one or another part of the error measure may be dominant.

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Section: The Two-phase Obstacle Problem In 1dmentioning

confidence: 65%

Verifications of Primal Energy Identities for Variational Problems with Obstacles

Repin

Valdman²

2018

Large-Scale Scientific Computing

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“…This result helps us in drawing significant qualitative conclusions about the optimal shapes. An effective numerical method is presented in [4].…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

An Elliptic Optimal Control Problem and its Two Relaxations

Emamizadeh

Farjudian

Mikayelyan

2016

J Optim Theory Appl

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In this note, we consider a control theory problem involving a strictly convex energy functional, which is not Gâteaux differentiable. The functional came up in the study of a shape optimization problem, and here we focus on the minimization of this functional. We relax the problem in two different ways, and show that the relaxed variants can be solved by applying some recent results on two-phase obstacle like problems of free boundary type. We derive an important qualitative property of the solutions, i. e., we prove that the minimizers are three-valued, a result which significantly reduces the search space for the relevant numerical algorithms.

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“…We verify the error identity (2.10) and the majorant estimate (3.12) on a benchmark example taken from [9]. In this example, Ω = (−1, 1), f = 0, α ⊕ = α = 8 and the equation is supplied with the Dirichlet boundary conditions u(−1) = −1, u(1) = 1.…”

Section: Examplementioning

confidence: 98%

A Posteriori Error Estimates for Two-Phase Obstacle Problem

Repin

Valdman²

2015

J Math Sci

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UDC 519.6For the two-phase obstacle problem we derive the basic error identity which yields natural measure of the distance to the exact solution. For this measure we derive a computable majorant valid for any function in the admissible (energy) class of functions. It is proved that the majorant vanishes if and only if the function coincides with the minimizer. It is shown that the respective estimate has no gap, so that accuracy of any approximation can be evaluated with any desired accuracy. Bibliography: 10 titles. Illustrations: 1 figure. Dedicated to dear Nina Nikolaevna Uraltseva in occasion of her jubilee 1 Two-Phase Obstacle Type ProblemLet Ω ⊂ R d (d 1) be a bounded domain with Lipschitz continuous boundary ∂Ω. We consider the following two-phase obstacle problem, which was introduced and studied in [1]-[3] and other papers cited therein.(1.1)

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Numerical solutions of a two-phase membrane problem

Cited by 13 publications

References 15 publications

Verifications of Primal Energy Identities for Variational Problems with Obstacles

Verifications of Primal Energy Identities for Variational Problems with Obstacles

An Elliptic Optimal Control Problem and its Two Relaxations

A Posteriori Error Estimates for Two-Phase Obstacle Problem

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