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A continuous finite element method with face penalty to approximate Friedrichs' systems

Erik Burman, Alexandre Ern (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

A continuous finite element method to approximate Friedrichs' systems is proposed and analyzed. Stability is achieved by penalizing the jumps across mesh interfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence rates in the graph norm and suboptimal of order ½ convergence rates in the L2-norm. A variant of the method specialized to Friedrichs' systems associated with elliptic PDE's in mixed form and reducing the number...

A mixed–FEM and BEM coupling for a three-dimensional eddy current problem

Salim Meddahi, Virginia Selgas (2003)

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

We study in this paper the electromagnetic field generated in a conductor by an alternating current density. The resulting interface problem (see Bossavit (1993)) between the metal and the dielectric medium is treated by a mixed–FEM and BEM coupling method. We prove that our BEM-FEM formulation is well posed and that it leads to a convergent Galerkin method.

A mixed–FEM and BEM coupling for a three-dimensional eddy current problem

Salim Meddahi, Virginia Selgas (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study in this paper the electromagnetic field generated in a conductor by an alternating current density. The resulting interface problem (see Bossavit (1993)) between the metal and the dielectric medium is treated by a mixed–FEM and BEM coupling method. We prove that our BEM-FEM formulation is well posed and that it leads to a convergent Galerkin method.

A numerical minimization scheme for the complex Helmholtz equation

Russell B. Richins, David C. Dobson (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We use the work of Milton, Seppecher, and Bouchitté on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate...

A numerical minimization scheme for the complex Helmholtz equation

Russell B. Richins, David C. Dobson (2012)

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

We use the work of Milton, Seppecher, and Bouchitté on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate...

A Slideing Mesh-Mortar Method for a two Dimensional Currents Model of Electric Engines

Annalisa Buffa, Yvon Maday, Francesca Rapetti (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The paper deals with the application of a non-conforming domain decomposition method to the problem of the computation of induced currents in electric engines with moving conductors. The eddy currents model is considered as a quasi-static approximation of Maxwell equations and we study its two-dimensional formulation with either the modified magnetic vector potential or the magnetic field as primary variable. Two discretizations are proposed, the first one based on curved finite elements and the...

A sliding Mesh-Mortar method for a two dimensional Eddy currents model of electric engines

Annalisa Buffa, Yvon Maday, Francesca Rapetti (2001)

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

The paper deals with the application of a non-conforming domain decomposition method to the problem of the computation of induced currents in electric engines with moving conductors. The eddy currents model is considered as a quasi-static approximation of Maxwell equations and we study its two-dimensional formulation with either the modified magnetic vector potential or the magnetic field as primary variable. Two discretizations are proposed, the first one based on curved finite elements and the...

An axisymmetric PIC code based on isogeometric analysis⋆

A. Back, A. Crestetto, A. Ratnani, E. Sonnendrücker (2011)

ESAIM: Proceedings

Isogeometric analysis has been developed recently to use basis functions resulting from the CAO description of the computational domain for the finite element spaces. The goal of this study is to develop an axisymmetric Finite Element PIC code in which specific spline Finite Elements are used to solve the Maxwell equations and the same spline functions serve as shape function for the particles. The computational domain itself is defined using splines...

An eddy current problem in terms of a time-primitive of the electric field with non-local source conditions

Alfredo Bermúdez, Bibiana López-Rodríguez, Rodolfo Rodríguez, Pilar Salgado (2013)

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

The aim of this paper is to analyze a formulation of the eddy current problem in terms of a time-primitive of the electric field in a bounded domain with input current intensities or voltage drops as source data. To this end, we introduce a Lagrange multiplier to impose the divergence-free condition in the dielectric domain. Thus, we obtain a time-dependent weak mixed formulation leading to a degenerate parabolic problem which we prove is well-posed. We propose a finite element method for space...

An electromagnetic damping machine: model, analysis and numerics

A. Buffa, Y. Maday, F. Rapetti (2001)

Bollettino dell'Unione Matematica Italiana

In questo lavoro viene considerato il modello bidimensionale completo di sistema elettromagnetico in movimento: le equazioni dei campi elettromagnetici sono accoppiate con quelle della meccanica e il sistema così ottenuto risulta essere non lineare nell'accoppiamento. Vengono analizzate la buona posizione del problema e la regolarità della soluzione continua; si propone inoltre uno schema di discretizzazione di tipo esplicito. Si dimostra la buona posizione e la convergenza della formulazione discreta...

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.

Computation of the demagnetizing potential in micromagnetics using a coupled finite and infinite elements method

François Alouges (2001)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is devoted to the practical computation of the magnetic potential induced by a distribution of magnetization in the theory of micromagnetics. The problem turns out to be a coupling of an interior and an exterior problem. The aim of this work is to describe a complete method that mixes the approaches of Ying [12] and Goldstein [6] which consists in constructing a mesh for the exterior domain composed of homothetic layers. It has the advantage of being well suited for catching the decay...

Computation of the demagnetizing potential in micromagnetics using a coupled finite and infinite elements method

François Alouges (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is devoted to the practical computation of the magnetic potential induced by a distribution of magnetization in the theory of micromagnetics. The problem turns out to be a coupling of an interior and an exterior problem. The aim of this work is to describe a complete method that mixes the approaches of Ying [12] and Goldstein [6] which consists in constructing a mesh for the exterior domain composed of homothetic layers. It has the advantage of being well suited for catching the...

Computing guided modes for an unbounded stratified medium in integrated optics

Fabrice Mahé (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a finite element method to compute guided modes in a stratified medium. The major difficulty to overcome is related to the unboundedness of the stratified medium. Our method is an alternative to the use of artificial boundary conditions and to the use of integral representation formulae. The domain is bounded in such a way we can write the solution on its lateral boundaries in terms of Fourier series. The series is then truncated for the computations over the bounded domain. The problem...

Computing guided modes for an unbounded stratified medium in integrated optics

Fabrice Mahé (2001)

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

We present a finite element method to compute guided modes in a stratified medium. The major difficulty to overcome is related to the unboundedness of the stratified medium. Our method is an alternative to the use of artificial boundary conditions and to the use of integral representation formulae. The domain is bounded in such a way we can write the solution on its lateral boundaries in terms of Fourier series. The series is then truncated for the computations over the bounded domain. The problem...

Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation*

Snorre H. Christiansen, Claire Scheid (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

As an example of a simple constrained geometric non-linear wave equation, we study a numerical approximation of the Maxwell Klein Gordon equation. We consider an existing constraint preserving semi-discrete scheme based on finite elements and prove its convergence in space dimension 2 for initial data of finite energy.

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