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Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

Eric Cancès, Rachida Chakir, Yvon Maday (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

In this article, we provide a priori error estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizsäcker (TFW) model and for the spectral discretization of the periodic Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the electronic ground state energy and density of molecular systems in the condensed phase. The TFW model is strictly convex with respect to the...

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 approach to a rate-independent model of decohesion in laminated composites

Zeman, Jan, Gruber, Pavel (2010)

Programs and Algorithms of Numerical Mathematics

In this paper, we present a numerical approach to evolution of decohesion in laminated composites based on incremental variational problems. An energy-based framework is adopted, in which we characterize the system by the stored energy and dissipation functionals quantifying reversible and irreversible processes, respectively. The time-discrete evolution then follows from a solution of incremental minimization problems, which are converted to a fully discrete form by employing the conforming finite...

Numerical approaches to rate-independent processes and applications in inelasticity

Alexander Mielke, Tomáš Roubíček (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

A conceptual numerical strategy for rate-independent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The novelty is that we obtain convergence of subsequences of space-time discretizations even in case where the limit problem does not have a unique solution and we need no additional assumptions on higher regularity of the limit solution. The variety of general perspectives thus obtained is illustrated on several...

Numerical approaches to the modelling of quasi-brittle crack propagation

Jiří Vala (2023)

Archivum Mathematicum

Computational analysis of quasi-brittle fracture in cement-based and similar composites, supplied by various types of rod, fibre, etc. reinforcement, is crucial for the prediction of their load bearing ability and durability, but rather difficult because of the risk of initiation of zones of microscopic defects, followed by formation and propagation of a large number of macroscopic cracks. A reasonable and complete deterministic description of relevant physical processes is rarely available. Thus,...

Numerical Approximation of a Fractional-In-Space Diffusion Equation (II) – with Nonhomogeneous Boundary Conditions

Ilic, M., Liu, F., Turner, I., Anh, V. (2006)

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: 26A33 (primary), 35S15In this paper, a space fractional diffusion equation (SFDE) with nonhomogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. 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 raised to the...

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 density dependent diffusion in age-structured population dynamics

Gerardo-Giorda, Luca (2013)

Applications of Mathematics 2013

We study a numerical method for the diffusion of an age-structured population in a spatial environment. We extend the method proposed in [2] for linear diffusion problem, to the nonlinear case, where the diffusion coefficients depend on the total population. We integrate separately the age and time variables by finite differences and we discretize the space variable by finite elements. We provide stability and convergence results and we illustrate our approach with some numerical result.

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.

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