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C++ Tools to construct our user-level language

Frédéric Hecht (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The aim of this paper is to present how to make a dedicaded computed language polymorphic and multi type, in C++ to solve partial differential equations with the finite element method. The driving idea is to make the language as close as possible to the mathematical notation.

C++ tools to construct our user-level language

Frédéric Hecht (2002)

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

The aim of this paper is to present how to make a dedicaded computed language polymorphic and multi type, in C++to solve partial differential equations with the finite element method. The driving idea is to make the language as close as possible to the mathematical notation.

Conservation schemes for convection-diffusion equations with Robin boundary conditions

Stéphane Flotron, Jacques Rappaz (2013)

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

In this article, we present a numerical scheme based on a finite element method in order to solve a time-dependent convection-diffusion equation problem and satisfy some conservation properties. In particular, our scheme is able to conserve the total energy for a heat equation or the total mass of a solute in a fluid for a concentration equation, even if the approximation of the velocity field is not completely divergence-free. We establish a priori errror estimates for this scheme and we give some...

Error estimates for the finite element discretization of semi-infinite elliptic optimal control problems

Pedro Merino, Ira Neitzel, Fredi Tröltzsch (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we derive a priori error estimates for linear-quadratic elliptic optimal control problems with finite dimensional control space and state constraints in the whole domain, which can be written as semi-infinite optimization problems. Numerical experiments are conducted to ilustrate our theory.

Finite element approximation of a Stefan problem with degenerate Joule heating

John W. Barrett, Robert Nürnberg (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a fully practical finite element approximation of the following degenerate system t ρ ( u ) - . ( α ( u ) u ) σ ( u ) | φ | 2 , . ( σ ( u ) φ ) = 0 subject to an initial condition on the temperature, u, and boundary conditions on both u and the electric potential, ϕ. In the above p(u) is the enthalpy incorporating the latent heat of melting, α(u) > 0 is the temperature dependent heat conductivity, and σ(u) > 0 is the electrical conductivity. The latter is zero in the frozen zone, u ≤ 0, which gives rise to the degeneracy in this Stefan...

Finite element approximation of a Stefan problem with degenerate Joule heating

John W. Barrett, Robert Nürnberg (2004)

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

We consider a fully practical finite element approximation of the following degenerate system t ρ ( u ) - . ( α ( u ) u ) σ ( u ) | φ | 2 , . ( σ ( u ) φ ) = 0 subject to an initial condition on the temperature, u , and boundary conditions on both u and the electric potential, φ . In the above ρ ( u ) is the enthalpy incorporating the latent heat of melting, α ( u ) > 0 is the temperature dependent heat conductivity, and σ ( u ) 0 is the electrical conductivity. The latter is zero in the frozen zone, u 0 , which gives rise to the degeneracy in this Stefan system. In addition to showing stability...

Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model

Francisco Guillén-González, Juan Vicente Gutiérrez-Santacreu (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We analyze two numerical schemes of Euler type in time and C0 finite-element type with 1 -approximation in space for solving a phase-field model of a binary alloy with thermal properties. This model is written as a highly non-linear parabolic system with three unknowns: phase-field, solute concentration and temperature, where the diffusion for the temperature and solute concentration may degenerate. The first scheme is nonlinear, unconditionally stable and convergent. The other scheme is linear...

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