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Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations*

Eduardo Casas, Fredi Tröltzsch (2011)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One of the difficulties in this analysis is that, in spite of the partial differential equation has a unique solution for any given control, the uniqueness...

Numerical analysis of the adiabatic variable method for the approximation of the nuclear hamiltonian

Yvon Maday, Gabriel Turinici (2001)

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

Many problems in quantum chemistry deal with the computation of fundamental or excited states of molecules and lead to the resolution of eigenvalue problems. One of the major difficulties in these computations lies in the very large dimension of the systems to be solved. Indeed these eigenfunctions depend on 3 n variables where n stands for the number of particles (electrons and/or nucleari) in the molecule. In order to diminish the size of the systems to be solved, the chemists have proposed many...

Numerical Analysis of the Adiabatic Variable Method for the Approximation of the Nuclear Hamiltonian

Yvon Maday, Gabriel Turinici (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Many problems in quantum chemistry deal with the computation of fundamental or excited states of molecules and lead to the resolution of eigenvalue problems. One of the major difficulties in these computations lies in the very large dimension of the systems to be solved. Indeed these eigenfunctions depend on 3n variables where n stands for the number of particles (electrons and/or nucleari) in the molecule. In order to diminish the size of the systems to be solved, the chemists have proposed...

Numerical analysis of the general biharmonic problem by the finite element method

Jiří Hřebíček (1982)

Aplikace matematiky

The present paper deals with solving the general biharmonic problem by the finite element method using curved triangular finit C 1 -elements introduced by Ženíšek. The effect of numerical integration is analysed in the case of mixed boundary conditions and sufficient conditions for the uniform V O h -ellipticity are found.

Numerical analysis of the MFS for certain harmonic problems

Yiorgos-Sokratis Smyrlis, Andreas Karageorghis (2004)

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

The Method of Fundamental Solutions (MFS) is a boundary-type meshless method for the solution of certain elliptic boundary value problems. In this work, we investigate the properties of the matrices that arise when the MFS is applied to the Dirichlet problem for Laplace’s equation in a disk. In particular, we study the behaviour of the eigenvalues of these matrices and the cases in which they vanish. Based on this, we propose a modified efficient numerical algorithm for the solution of the problem...

Numerical analysis of the MFS for certain harmonic problems

Yiorgos-Sokratis Smyrlis, Andreas Karageorghis (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The Method of Fundamental Solutions (MFS) is a boundary-type meshless method for the solution of certain elliptic boundary value problems. In this work, we investigate the properties of the matrices that arise when the MFS is applied to the Dirichlet problem for Laplace's equation in a disk. In particular, we study the behaviour of the eigenvalues of these matrices and the cases in which they vanish. Based on this, we propose a modified efficient numerical algorithm for the solution of the problem...

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

Eric Cancès, Rachida Chakir, Yvon Maday (2012)

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

In this article, we provide a priorierror 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 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 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 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 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 approximation of the inviscid 3D primitive equations in a limited domain

Qingshan Chen, Ming-Cheng Shiue, Roger Temam, Joseph Tribbia (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

A new set of nonlocal boundary conditions is proposed for the higher modes of the 3D inviscid primitive equations. Numerical schemes using the splitting-up method are proposed for these modes. Numerical simulations of the full nonlinear primitive equations are performed on a nested set of domains, and the results are discussed.

Numerical approximation of the inviscid 3D primitive equations in a limited domain

Qingshan Chen, Ming-Cheng Shiue, Roger Temam, Joseph Tribbia (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

A new set of nonlocal boundary conditions is proposed for the higher modes of the 3D inviscid primitive equations. Numerical schemes using the splitting-up method are proposed for these modes. Numerical simulations of the full nonlinear primitive equations are performed on a nested set of domains, and the results are discussed.

Currently displaying 481 – 500 of 603