87,002 research outputs found
Analysis of finite element approximation of evolution problems in mixed form
No full textThis paper deals with the finite element approximation of evolution problems in mixed form. Following [D. Boffi, F. Brezzi, L. Gastaldi, Math. Comp. 69 (2000), pp.121-140], we handle separately two types of problems. A model for the first case is the heat equation in mixed form, while the time dependent Stokes problem fits within the second one. For either case, we give sufficient conditions for a good approximation in the natural functional spaces. The results are not obvious in the first situation. In this case, the well-known conditions for the well-posedness and convergence of the corresponding steady problem are not sufficient for the good approximation of the time dependent problem. This issue is demonstrated with a numerical (counter-) example and justified analytically
Ceci n'est pas une utopie
No full textLe utopie sono tali finché rimangono indecidibili, e perciò irrealizzate. Sono dispositivi modali concretamente misti: mente e paradigma, immagine e rappresentazione, proiezione tecnico-figurale e volitivo-affettiva, immaginazione geografica e immaginazione politica, disegno e progetto, forma astratta e funzione storica concreta, finalità ed effettività, razionalità, organizzazione, prefigurazione, operatività. Utopia è una certa modalità ontologica imperfetta, portatrice di un’interrogazione critico-ironica dell’esistenza, basata sul paradosso e l’incongruente. Di tale natura è l'Utopia di Thomas More
Mixed finite elements for Maxwell's eigenproblem: the question of spurious modes
No full textH.G. Bock et al., eds
Distributed Lagrange Multiplier for Fluid-Structure Interactions
No full textInternational audienceIn this paper we make preliminary numerical tests to assess the performance of the scheme introduced in Boffi et al. (SIAM J Numer Anal 53(6):2584–2604, 2015) and analyzed in Boffi and Gastaldi (Numer Math 135(3):711–732, 2017) for the approximation of fluid-structure interaction problems. We show how to implement the scheme within the FreeFem++ framework (Hecht, J Numer Math 20(3–4):251–265, 2012) and validate our code with respect to some known problems and benchmarks. The main conclusion is that a simple implementation can provide quite accurate results for non trivial applications
First order least-squares formulations for eigenvalue problems
No full textIn this paper we discuss spectral properties of operators associated with the least-squares finite-element approximation of elliptic partial differential equations. The convergence of the discrete eigenvalues and eigenfunctions towards the corresponding continuous eigenmodes is studied and analyzed with the help of appropriate error estimates. A priori and a posteriori estimates are proved
Least-squares formulations for eigenvalue problems associated with linear elasticity
No full textWe study the approximation of the spectrum of least-squares operators arising from linear elasticity. We consider a two-field (stress/displacement) and a three-field (stress/displacement/vorticity) formulation; other formulations might be analyzed with similar techniques. We prove a priori estimates and we confirm the theoretical results with simple two-dimensional numerical experiments
On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form
No full textAbstract. In the approximation of linear elliptic operators in mixed form, it is well known that the so-called inf-sup and ellipticity in the kernel prop-erties are sufficient (and, in a sense to be made precise, necessary) in order to have good approximation properties and optimal error bounds. One might think, in the spirit of Mercier-Osborn-Rappaz-Raviart and in consideration of the good behavior of commonly used mixed elements (like Raviart–Thomas or Brezzi–Douglas–Marini elements), that these conditions are also sufficient to ensure good convergence properties for eigenvalues. In this paper we show that this is not the case. In particular we present examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues. Such bad behavior is proved analytically and demonstrated in nu-merical experiments. We also present additional assumptions (fulfilled by the commonly used mixed methods already mentioned) which guarantee optimal error bounds for eigenvalue approximations as well. 1
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