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A finite element method on composite grids based on Nitsche's method

Anita Hansbo, Peter Hansbo, Mats G. Larson (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we propose a finite element method for the approximation of second order elliptic problems on composite grids. The method is based on continuous piecewise polynomial approximation on each grid and weak enforcement of the proper continuity at an artificial interface defined by edges (or faces) of one the grids. We prove optimal order a priori and energy type a posteriori error estimates in 2 and 3 space dimensions, and present some numerical examples.

A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

Komla Domelevo, Pascal Omnes (2005)

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

We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires “Voronoi-type” meshes. We show the equivalence of this finite volume...

A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

Komla Domelevo, Pascal Omnes (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires “Voronoi-type” meshes. We show the equivalence of this finite volume...

A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients

Chokri Chniti (2011)

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

In this paper we certify that the same approach proposed in previous works by Chniti et al. [C. R. Acad. Sci. 342 (2006) 883–886; CALCOLO 45 (2008) 111–147; J. Sci. Comput. 38 (2009) 207–228] can be applied to more general operators with strong heterogeneity in the coefficients. We consider here the case of reaction-diffusion problems with piecewise constant coefficients. The problem reduces to determining the coefficients of some transmission conditions to obtain fast convergence of domain decomposition...

A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients

Chokri Chniti (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we certify that the same approach proposed in previous works by Chniti et al. [C. R. Acad. Sci.342 (2006) 883–886; CALCOLO45 (2008) 111–147; J. Sci. Comput.38 (2009) 207–228] can be applied to more general operators with strong heterogeneity in the coefficients. We consider here the case of reaction-diffusion problems with piecewise constant coefficients. The problem reduces to determining the coefficients of some transmission conditions to obtain fast convergence of domain decomposition...

A mixed-FEM and BEM coupling for the approximation of the scattering of thermal waves in locally non-homogeneous media

María-Luisa Rapún, Francisco-Javier Sayas (2006)

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

This paper proposes and analyzes a BEM-FEM scheme to approximate a time-harmonic diffusion problem in the plane with non-constant coefficients in a bounded area. The model is set as a Helmholtz transmission problem with adsorption and with non-constant coefficients in a bounded domain. We reformulate the problem as a four-field system. For the temperature and the heat flux we use piecewise constant functions and lowest order Raviart-Thomas elements associated to a triangulation approximating the...

A mixed-FEM and BEM coupling for the approximation of the scattering of thermal waves in locally non-homogeneous media

María-Luisa Rapún, Francisco-Javier Sayas (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper proposes and analyzes a BEM-FEM scheme to approximate a time-harmonic diffusion problem in the plane with non-constant coefficients in a bounded area. The model is set as a Helmholtz transmission problem with adsorption and with non-constant coefficients in a bounded domain. We reformulate the problem as a four-field system. For the temperature and the heat flux we use piecewise constant functions and lowest order Raviart-Thomas elements associated to a triangulation approximating the...

A multiscale correction method for local singular perturbations of the boundary

Marc Dambrine, Grégory Vial (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work, we consider singular perturbations of the boundary of a smooth domain. We describe the asymptotic behavior of the solution uE of a second order elliptic equation posed in the perturbed domain with respect to the size parameter ε of the deformation. We are also interested in the variations of the energy functional. We propose a numerical method for the approximation of uE based on a multiscale superposition of the unperturbed solution u0 and a profile defined in a model domain. We...

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...

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