A Quasi-Variational Inequality Problem in Superconductivity

Barrett, John W.; Prigozhin, Leonid

doi:10.1142/s0218202510004404

Cited by 51 publications

(85 citation statements)

References 21 publications

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“…Here we will use a mixed variational formulation of the growing sandpile model involving both variables. Such formulations are often advantageous, because they allow one to determine not only the evolving sand surface w but also the surface flux q, which is of interest too in various applications; see Prigozhin [20,21], and Barrett and Prigozhin [4]. In such formulations, and this is their additional advantage, the difficult to deal with gradient constraint in (1.6) is replaced by a simpler, although non-smooth, nonlinearity.…”

Section: Mathematical Models and Their Mixed Formulations (I) Growingmentioning

confidence: 99%

Sandpiles and superconductors: nonconforming linear finite element approximations for mixed formulations of quasi-variational inequalities

Barrett

Prigozhin²

2013

IMA Journal of Numerical Analysis

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Abstract. Similar evolutionary variational and quasi-variational inequalities with gradient constraints arise in the modeling of growing sandpiles and type-II superconductors. Recently, mixed formulations of these inequalities were used for establishing existence results in the quasi-variational inequality case. Such formulations, and this is an additional advantage, made it possible to determine numerically not only the primal variables, e.g. the evolving sand surface and the magnetic field for sandpiles and superconductors, respectively, but also the dual variables, the sand flux and the electric field.Numerical approximations of these mixed formulations in previous works employed the RaviartThomas element of the lowest order. Here we introduce simpler numerical approximations of these mixed formulations based on the nonconforming linear finite element. We prove (subsequence) convergence of these approximations, and illustrate their effectiveness by numerical experiments.

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Section: Mathematical Models and Their Mixed Formulations (I) Growingmentioning

confidence: 99%

Sandpiles and superconductors: nonconforming linear finite element approximations for mixed formulations of quasi-variational inequalities

Barrett

Prigozhin²

2013

IMA Journal of Numerical Analysis

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“…This is a natural formulation for general first order quasilinear scalar operators under constraint (2) and it leads to a new class of problems that were not considered in the classical references of quasivariational inequalities [3] or [7]. However, it contains as a special case the problem considered in [5], motivated by a criticalstate model in superconductivity. It was also considered in the setting of parabolic operators in [28,29], in the variational inequality case, that is, when G does not depend on the solution u, and in a quasivariational case by the authors in [26].…”

Section: Introductionmentioning

confidence: 99%

Quasivariational Solutions for First Order Quasilinear Equations with Gradient Constraint

Rodrigues

Santos

2012

Arch Rational Mech Anal

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We prove the existence of solutions for a quasi-variational inequality of evolution with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable a priori estimates. We also obtain the existence of stationary solutions by studying the asymptotic behaviour in time. In the variational case, corresponding to a constraint independent of the solution, we also give uniqueness results.

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“…For obtaining a system with nonzero right hand side f and homogeneous Dirichlet boundary conditions, as studied in this work, we extend g to g ext such that g ext|∂Ω = g, reformulate the system in [24] and solve (P). In our tests we choose g ext such that ∆g ext (x) =: f (x) = 800x [4]. In fact setting y = y…”

Section: Test Problem P3mentioning

confidence: 99%

Several path-following methods for a class of gradient constrained variational inequalities

Hintermüller

Rasch

2015

Computers & Mathematics with Applications

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Abstract. Path-following splitting and semismooth Newton methods for solving a class of problems related to elasto-plastic material deformations are proposed, analyzed and tested numerically. While the splitting techniques result in alternating minimization schemes, which are typically linearly convergent, the proposed Moreau-Yosida regularization based semismooth Newton technique and an associated lifting step yield local superlinear convergence in function space. The lifting step accounts for the fact that the operator associated with the linear system in the Newton iteration need not be boundedly invertible (uniformly along the iterates). For devising an efficient update strategy for the path-following parameter regularity properties of the path are studied and utilized within an inexact path-following scheme for all approaches. The paper ends by a report on numerical tests of the different approaches.

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A Quasi-Variational Inequality Problem in Superconductivity

Cited by 51 publications

References 21 publications

Sandpiles and superconductors: nonconforming linear finite element approximations for mixed formulations of quasi-variational inequalities

Sandpiles and superconductors: nonconforming linear finite element approximations for mixed formulations of quasi-variational inequalities

Quasivariational Solutions for First Order Quasilinear Equations with Gradient Constraint

Several path-following methods for a class of gradient constrained variational inequalities

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