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Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes

Martin Vohralík — 2006

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the lowest-order Raviart–Thomas mixed finite element method for second-order elliptic problems on simplicial meshes in two and three space dimensions. This method produces saddle-point problems for scalar and flux unknowns. We show how to easily and locally eliminate the flux unknowns, which implies the equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. The matrix of the final linear system is sparse, positive...

A finite element discretization of the contact between two membranes

Faker Ben BelgacemChristine BernardiAdel BlouzaMartin Vohralík — 2009

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

From the fundamental laws of elasticity, we write a model for the contact between two membranes and we perform the analysis of the corresponding system of variational inequalities. We propose a finite element discretization of this problem and prove its well-posedness. We also establish a priori and a posteriori error estimates.

Guaranteed and robust error estimates for singularly perturbed reaction–diffusion problems

Ibrahim CheddadiRadek FučíkMariana I. PrietoMartin Vohralík — 2009

ESAIM: Mathematical Modelling and Numerical Analysis

We derive error estimates for singularly perturbed reaction–diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a generic constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general...

A finite element discretization of the contact between two membranes

Faker Ben BelgacemChristine BernardiAdel BlouzaMartin Vohralík — 2008

ESAIM: Mathematical Modelling and Numerical Analysis

From the fundamental laws of elasticity, we write a model for the contact between two membranes and we perform the analysis of the corresponding system of variational inequalities. We propose a finite element discretization of this problem and prove its well-posedness. We also establish and error estimates.

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