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Proposition de préconditionneurs pseudo-différentiels pour l’équation CFIE de l’électromagnétisme

David P. Levadoux (2005)

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

We present a weak parametrix of the operator of the CFIE equation. An interesting feature of this parametrix is that it is compatible with different discretization strategies and hence allows for the construction of efficient preconditioners dedicated to the CFIE. Furthermore, one shows that the underlying operator of the CFIE verifies an uniform discrete Inf-Sup condition which allows to predict an original convergence result of the numerical solution of the CFIE to the exact one.

Proposition de préconditionneurs pseudo-différentiels pour l'équation CFIE de l'électromagnétisme

David P. Levadoux (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

On exhibe dans cette note une paramétrix (au sens faible) de l'opérateur sous-jacent à l'équation CFIE de l'électromagnétisme. L'intérêt de cette paramétrix est de se prêter à différentes stratégies de discrétisation et ainsi de pouvoir être utilisée comme préconditionneur de la CFIE. On montre aussi que l'opérateur sous-jacent à la CFIE satisfait une condition Inf-Sup discrète uniforme, applicable aux espaces de discrétisation usuellement rencontrés en électromagnétisme, et qui permet d'établir...

Recent developments in wavelet methods for the solution of PDE's

Silvia Bertoluzza (2005)

Bollettino dell'Unione Matematica Italiana

After reviewing some of the properties of wavelet bases, and in particular the property of characterisation of function spaces via wavelet coefficients, we describe two new approaches to, respectively, stabilisation of numerically unstable PDE's and to non linear (adaptive) solution of PDE's, which are made possible by these properties.

Replicant compression coding in Besov spaces

Gérard Kerkyacharian, Dominique Picard (2010)

ESAIM: Probability and Statistics

We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space B π , q s on a regular domain of ℝ d . The result is: if s - d(1/π - 1/p)+> 0, then the Kolmogorov metric entropy satisfies H(ε) ~ ε-d/s. This proof takes advantage of the representation of such spaces on wavelet type bases and extends the result to more general spaces. The lower bound is a consequence of very simple probabilistic exponential inequalities. To prove the upper bound,...

Replicant compression coding in Besov spaces

Gérard Kerkyacharian, Dominique Picard (2003)

ESAIM: Probability and Statistics

We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space B π , q s on a regular domain of ℝ d . The result is: if s - d ( 1 / π - 1 / p ) + > 0 , then the Kolmogorov metric entropy satisfies H ( ϵ ) ∼ ϵ - d / s . This...

Robust operator estimates and the application to substructuring methods for first-order systems

Christian Wieners, Barbara Wohlmuth (2014)

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

We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite general partial differential operators. The starting point of our analysis is the DPG method introduced by [Demkowicz et al., SIAM J. Numer. Anal. 49 (2011) 1788–1809; Zitelli et al., J. Comput. Phys. 230 (2011) 2406–2432]. This discretization results in a sparse positive definite linear algebraic system which can be obtained from a saddle point problem by an element-wise Schur complement reduction applied to the test space....

Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case

Paola F. Antonietti, Blanca Ayuso (2007)

ESAIM: Mathematical Modelling and Numerical Analysis


We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain...

Semiregular finite elements in solving some nonlinear problems

Jana Zlámalová (2001)

Applications of Mathematics

In this paper, under the maximum angle condition, the finite element method is analyzed for nonlinear elliptic variational problem formulated in [4]. In [4] the analysis was done under the minimum angle condition.

Semiregular hermite tetrahedral finite elements

Alexander Ženíšek, Jana Hoderová-Zlámalová (2001)

Applications of Mathematics

Tetrahedral finite C 0 -elements of the Hermite type satisfying the maximum angle condition are presented and the corresponding finite element interpolation theorems in the maximum norm are proved.

Simplicial finite elements in higher dimensions

Jan Brandts, Sergey Korotov, Michal Křížek (2007)

Applications of Mathematics

Over the past fifty years, finite element methods for the approximation of solutions of partial differential equations (PDEs) have become a powerful and reliable tool. Theoretically, these methods are not restricted to PDEs formulated on physical domains up to dimension three. Although at present there does not seem to be a very high practical demand for finite element methods that use higher dimensional simplicial partitions, there are some advantages in studying the methods independent of the...

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