Existence and approximation of a mixed formulation for thin film magnetization problems in superconductivity

2014 ◽  
Vol 24 (05) ◽  
pp. 991-1015 ◽  
Author(s):  
John W. Barrett ◽  
Leonid Prigozhin
Keyword(s):  
Thin Film ◽  
Electric Field ◽  
Finite Element Approximation ◽  
Mixed Formulation ◽  
Approximation Formula ◽  
Element Approximation ◽  
Type Ii Superconductors ◽  
Current Voltage ◽  
Current Voltage Relation ◽  
Film Magnetization

We recall a recently introduced mixed formulation of thin film magnetization problems for type-II superconductors written in terms of two variables, the electric field and the magnetization function, see [Electric field formulation for thin film magnetization problems, Supercond. Sci. Technol.25 (2012) 104002]. A finite element approximation, [Formula: see text], based on this mixed formulation, involving the lowest-order Raviart–Thomas element for approximating the electric field, was also introduced in [Electric field formulation for thin film magnetization problems, Supercond. Sci. Technol.25 (2012) 104002]. Here h, τ are the spatial and temporal discretization parameters, and [Formula: see text] with p-1 the value of power in the current–voltage relation characterizing the superconducting material. In this paper, we establish well-posedness of [Formula: see text], and prove convergence of the unique solution of [Formula: see text] to a solution of the power law model ( Q r), for a fixed r > 1, as h, τ → 0. In addition, we prove convergence of a solution of ( Q r) to a solution of the critical state model (Q), as r → 1. Hence, we prove existence of solutions to ( Q r), for a fixed r > 1, and (Q). Finally, numerical experiments are presented.

scholarly journals Approximation Technics for an Unsteady Dynamic Koiter Shell

2007 ◽  
Vol 2007 ◽  
pp. 1-15
Author(s):  
Saloua Mani Aouadi
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Mixed Formulation ◽  
Element Approximation

We propose a mixed formulation in dynamical elasticity of shells which allows a locking-free finite element approximation in particular cases of Koiter shells.

scholarly journals ANALYSIS THE EFFECTS OF FERROELECTRIC ELECTRIC FIELD ON CURRENT-VOLTAGE CHARACTERISTICS OF THE FERROELECTRIC FIELD EFFECT TRANSISTOR USING SRBI2TA2O9 THIN FILM

2020 ◽  
Vol 28 ◽  
Author(s):  
AN NGUYEN VAN ◽  
AN VO XUAN ◽  
HONG THAM LE THI
Keyword(s):  
Thin Film ◽  
Electric Field ◽  
Field Effect ◽  
Field Effect Transistor ◽  
Current Voltage ◽  
Current Voltage Characteristics ◽  
Effect Transistor

ANALYSIS THE EFFECTS OF FERROELECTRIC ELECTRIC FIELD ONCURRENT-VOLTAGE CHARACTERISTICS OF THE FERROELECTRICFIELD EFFECT TRANSISTOR USING SRBI2TA2O9 THIN FILM

scholarly journals Finite Element Approximation of Surfactant Spreading on a Thin Film

2003 ◽  
Vol 41 (4) ◽  
pp. 1427-1464 ◽  
Author(s):  
John W. Barrett ◽  
Harald Garcke ◽  
Robert Nürnberg
Keyword(s):  
Thin Film ◽  
Finite Element ◽  
Finite Element Approximation ◽  
Element Approximation

scholarly journals Finite Element Approximation of Soluble Surfactant Spreading on a Thin Film

2006 ◽  
Vol 44 (3) ◽  
pp. 1218-1247 ◽  
Author(s):  
John W. Barrett ◽  
Robert Nürnberg ◽  
Mark R. E. Warner
Keyword(s):  
Thin Film ◽  
Finite Element ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Soluble Surfactant

Stable finite element approximation of a Cahn–Hilliard–Stokes system coupled to an electric field

2016 ◽  
Vol 28 (3) ◽  
pp. 470-498
Author(s):  
ROBERT NÜRNBERG ◽  
EDWARD J. W. TUCKER
Keyword(s):  
Finite Element ◽  
Electric Field ◽  
Organic Solar Cells ◽  
Stokes System ◽  
Chemical Potential ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Finite Element Approximation ◽  
Strength Parameter ◽  
Element Approximation

We consider a fully practical finite element approximation of the Cahn–Hilliard–Stokes system: $$\begin{align*} \gamma \tfrac{\partial u}{\partial t} + \beta v \cdot \nabla u - \nabla \cdot \left( \nabla w \right) & = 0 \,, \quad w= -\gamma \Delta u + \gamma ^{-1} \Psi ' (u) - \tfrac12 \alpha c'(\cdot,u) | \nabla \phi |^2\,, \\ \nabla \cdot (c(\cdot,u) \nabla \phi) & = 0\,,\quad \begin{cases} -\Delta v + \nabla p = \varsigma w \nabla u, \\ \nabla \cdot v = 0, \end{cases} \end{align*}$$ subject to an initial condition u0(.) ∈ [−1, 1] on the conserved order parameter u ∈ [−1, 1], and mixed boundary conditions. Here, γ ∈ $\mathbb{R}_{>0}$ is the interfacial parameter, α ∈ $\mathbb{R}_{\geq0}$ is the field strength parameter, Ψ is the obstacle potential, c(⋅, u) is the diffusion coefficient, and c′(⋅, u) denotes differentiation with respect to the second argument. Furthermore, w is the chemical potential, φ is the electro-static potential, and (v, p) are the velocity and pressure. The system has been proposed to model the manipulation of morphologies in organic solar cells with the help of an applied electric field and kinetics.

scholarly journals On the ionospheric coupling of auroral electric fields

2009 ◽  
Vol 16 (2) ◽  
pp. 365-372 ◽  
Author(s):  
G. T. Marklund
Keyword(s):  
Electric Field ◽  
Electric Fields ◽  
Small Scale ◽  
Scale Size ◽  
Current Voltage ◽  
Current Region ◽  
Scale Structures ◽  
Downward Currents ◽  
Current Voltage Relation ◽  
Small Scale Structures

Abstract. The quasi-static coupling of high-altitude potential structures and electric fields to the ionosphere is discussed with particular focus on the downward field-aligned current (FAC) region. Results are presented from a preliminary analysis of a selection of electric field events observed by Cluster above the acceleration region. The degree of coupling is here estimated as the ratio between the magnetic field-aligned potential drop, ΔΦII, as inferred from the characteristic energy of upward ion (electron) beams for the upward (downward) current region and the high-altitude perpendicular (to B) potential, ΔΦbot, as calculated by integrating the perpendicular electric field across the structure. For upward currents, the coupling can be expressed analytically, using the linear current-voltage relation, as outlined by Weimer et al. (1985). This gives a scale size dependent coupling where structures are coupled (decoupled) above (below) a critical scale size. For downward currents, the current-voltage relation is highly non-linear which complicates the understanding of how the coupling works. Results from this experimental study indicate that small-scale structures are decoupled, similar to small-scale structures in the upward current region. There are, however, exceptions to this rule as illustrated by Cluster results of small-scale intense electric fields, correlated with downward currents, indicating a perfect coupling between the ionosphere and Cluster altitude.

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