Displaying similar documents to “A Superconvergence result for mixed finite element approximations of the eigenvalue problem”

A Superconvergence result for mixed finite element approximations of the eigenvalue problem

Qun Lin, Hehu Xie (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

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In this paper, we present a superconvergence result for the mixed finite element approximations of general second order elliptic eigenvalue problems. It is known that a superconvergence result has been given by Durán [ (1999) 1165–1178] and Gardini [ (2009) 853–865] for the lowest order Raviart-Thomas approximation of Laplace eigenvalue problems. In this work, we introduce a new way to derive the superconvergence of general second order elliptic eigenvalue...

A non elliptic spectral problem related to the analysis of superconducting micro-strip lines

Anne-Sophie Bonnet-Bendhia, Karim Ramdani (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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This paper is devoted to the spectral analysis of a non elliptic operator , deriving from the study of superconducting micro-strip lines. Once a sufficient condition for the self-adjointness of operator has been derived, we determine its continuous spectrum. Then, we show that is unbounded from below and that it has a sequence of negative eigenvalues tending to -∞. Using the Min-Max principle, a characterization of its positive eigenvalues is given. Thanks to this characterization,...

A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity

Yongxing Shen, Adrian J. Lew (2012)

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

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We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for nearly and perfectly incompressible linear elasticity. These mixed methods allow the choice of polynomials of any order  ≥ 1 for the approximation of the displacement field, and of order or  − 1 for the pressure space, and are stable for any positive value of the stabilization parameter. We prove the optimal convergence of the displacement and stress fields in both cases, with error estimates that...

Finite volume schemes for fully non-linear elliptic equations in divergence form

Jérôme Droniou (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

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We construct finite volume schemes, on unstructured and irregular grids and in any space dimension, for non-linear elliptic equations of the -Laplacian kind: -div(|∇∇) = ƒ (with 1 < < ∞). We prove the existence and uniqueness of the approximate solutions, as well as their strong convergence towards the solution of the PDE. The outcome of some numerical tests are also provided.

Basic principles of mixed Virtual Element Methods

F. Brezzi, Richard S. Falk, L. Donatella Marini (2014)

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

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The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of (div)-conforming vector fields (or, more generally, of ( − 1) − ). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the...

Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin laplacian

Pedro Ricardo Simão Antunes, Pedro Freitas, James Bernard Kennedy (2013)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider the problem of minimising the th-eigenvalue of the Robin Laplacian in R. Although for  = 1,2 and a positive boundary parameter it is known that the minimisers do not depend on , we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on . We derive a Wolf–Keller type result for this problem and show that optimal eigenvalues grow at most with , which is in sharp contrast with the Weyl asymptotics for a...

On torsional rigidity and principal frequencies: an invitation to the Kohler−Jobin rearrangement technique

Lorenzo Brasco (2014)

ESAIM: Control, Optimisation and Calculus of Variations

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We generalize to the -Laplacian a spectral inequality proved by M.-T. Kohler−Jobin. As a particular case of such a generalization, we obtain a sharp lower bound on the first Dirichlet eigenvalue of of a set in terms of its -torsional rigidity. The result is valid in every space dimension, for every 1    ∞ and for every open set with finite measure. Moreover, it holds by replacing the first eigenvalue with more general optimal Poincaré-Sobolev constants....

Approximation of solution branches for semilinear bifurcation problems

Laurence Cherfils (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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This note deals with the approximation, by a finite element method with numerical integration, of solution curves of a semilinear problem. Because of both mixed boundary conditions and geometrical properties of the domain, some of the solutions do not belong to . So, classical results for convergence lead to poor estimates. We show how to improve such estimates with the use of weighted Sobolev spaces together with a mesh “ adapted” to the singularity....