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Numerical analysis of the quasistatic thermoviscoelastic thermistor problem

José R. Fernández (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work, the quasistatic thermoviscoelastic thermistor problem is considered. The thermistor model describes the combination of the effects due to the heat, electrical current conduction and Joule's heat generation. The variational formulation leads to a coupled system of nonlinear variational equations for which the existence of a weak solution is recalled. Then, a fully discrete algorithm is introduced based on the finite element method to approximate the spatial variable and an Euler scheme...

Numerical Approximation of a Fractional-In-Space Diffusion Equation, I

Ilic, M., Liu, F., Turner, I., Anh, V. (2005)

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: 26A33 (primary), 35S15 (secondary)This paper provides a new method and corresponding numerical schemes to approximate a fractional-in-space diffusion equation on a bounded domain under boundary conditions of the Dirichlet, Neumann or Robin type. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix...

Numerical approximation of dynamic deformations of a thermoviscoelastic rod against an elastic obstacle

Maria I. M. Copetti (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we consider a hyperbolic-parabolic problem that models the longitudinal deformations of a thermoviscoelastic rod supported unilaterally by an elastic obstacle. The existence and uniqueness of a strong solution is shown. A finite element approximation is proposed and its convergence is proved. Numerical experiments are reported.

Numerical approximation of dynamic deformations of a thermoviscoelastic rod against an elastic obstacle

Maria I.M. Copetti (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we consider a hyperbolic-parabolic problem that models the longitudinal deformations of a thermoviscoelastic rod supported unilaterally by an elastic obstacle. The existence and uniqueness of a strong solution is shown. A finite element approximation is proposed and its convergence is proved. Numerical experiments are reported.

Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations

Antoine Gloria (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice ℤ d with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with...

Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations

Antoine Gloria (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice ℤ d with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with...

Numerical Approximations of the Relative Rearrangement: The piecewise linear case. Application to some Nonlocal Problems

Jean-Michel Rakotoson, Maria Luisa Seoane (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We first prove an abstract result for a class of nonlocal problems using fixed point method. We apply this result to equations revelant from plasma physic problems. These equations contain terms like monotone or relative rearrangement of functions. So, we start the approximation study by using finite element to discretize this nonstandard quantities. We end the paper by giving a numerical resolution of a model containing those terms.

Numerical computation of solitons for optical systems

Laurent Di Menza (2009)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number k of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large k are large nonlinear exponents σ . In a second part, we compute...

Numerical computation of solitons for optical systems

Laurent Di Menza (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number k of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large k are large nonlinear exponents σ. In a second part, we compute...

Numerical homogenization for indefinite H(curl)-problems

Verfürth, Barbara (2017)

Proceedings of Equadiff 14

In this paper, we present a numerical homogenization scheme for indefinite, timeharmonic Maxwell’s equations involving potentially rough (rapidly oscillating) coefficients. The method involves an H(curl)-stable, quasi-local operator, which allows for a correction of coarse finite element functions such that order optimal (w.r.t. the mesh size) error estimates are obtained. To that end, we extend the procedure of [D. Gallistl, P. Henning, B. Verfürth, Numerical homogenization for H(curl)-problems,...

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