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An optimal error bound for a finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy

John W. Barrett, James F. Blowey (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Using the approach in [5] for analysing time discretization error and assuming more regularity on the initial data, we improve on the error bound derived in [2] for a fully practical piecewise linear finite element approximation with a backward Euler time discretization of a model for phase separation of a multi-component alloy with non-smooth free energy.

Analyse de sensibilité d’un problème de contrôle optimal bilinéaire

Jean-Marc Clérin (2012)

Annales mathématiques Blaise Pascal

Dans cet article, nous étudions la sensibilité d’un problème de contrôle optimal de type bilinéaire. Le coût est différentiable, quadratique et strictement convexe. Le système est gouverné par un opérateur parabolique du quatrième ordre et présente une perturbation additive dans l’équation d’état, ainsi qu’une partie bilinéaire, relativement au contrôle u et à l’état z , de la forme ( u · ) z . Sous des conditions de petitesse de l’état initial et de la perturbation, nous exploitons les propriétés de régularité...

Asymptotic behavior of a sixth-order Cahn-Hilliard system

Alain Miranville (2014)

Open Mathematics

Our aim in this paper is to study the asymptotic behavior, in terms of finite-dimensional attractors, of a sixth-order Cahn-Hilliard system. This system is based on a modification of the Ginzburg-Landau free energy proposed in [Torabi S., Lowengrub J., Voigt A., Wise S., A new phase-field model for strongly anisotropic systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2009, 465(2105), 1337–1359], assuming isotropy.

Classical global solutions of the initial boundary value problems for a class of nonlinear parabolic equations

Guo Wang Chen (1994)

Commentationes Mathematicae Universitatis Carolinae

The existence, uniqueness and regularities of the generalized global solutions and classical global solutions to the equation u t = - A ( t ) u x 4 + B ( t ) u x 2 + g ( u ) x 2 + f ( u ) x + h ( u x ) x + G ( u ) with the initial boundary value conditions u ( - , t ) = u ( , t ) = 0 , u x 2 ( - , t ) = u x 2 ( , t ) = 0 , u ( x , 0 ) = ϕ ( x ) , or with the initial boundary value conditions u x ( - , t ) = u x ( , t ) = 0 , u x 3 ( - , t ) = u x 3 ( , t ) = 0 , u ( x , 0 ) = ϕ ( x ) , are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions.

Control of the wave equation by time-dependent coefficient

Antonin Chambolle, Fadil Santosa (2002)

ESAIM: Control, Optimisation and Calculus of Variations

We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior...

Control of the Wave Equation by Time-Dependent Coefficient

Antonin Chambolle, Fadil Santosa (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior of...

Convergent semidiscretization of a nonlinear fourth order parabolic system

Ansgar Jüngel, René Pinnau (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.

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