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Electromagnetic scattering at composite objects : a novel multi-trace boundary integral formulation

Xavier Claeys, Ralf Hiptmair (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

Since matrix compression has paved the way for discretizing the boundary integral equation formulations of electromagnetics scattering on very fine meshes, preconditioners for the resulting linear systems have become key to efficient simulations. Operator preconditioning based on Calderón identities has proved to be a powerful device for devising preconditioners. However, this is not possible for the usual first-kind boundary formulations for electromagnetic...

Electromagnetic scattering at composite objects : a novel multi-trace boundary integral formulation

Xavier Claeys, Ralf Hiptmair (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

Since matrix compression has paved the way for discretizing the boundary integral equation formulations of electromagnetics scattering on very fine meshes, preconditioners for the resulting linear systems have become key to efficient simulations. Operator preconditioning based on Calderón identities has proved to be a powerful device for devising preconditioners. However, this is not possible for the usual first-kind boundary formulations for electromagnetic...

Elementary linear algebra for advanced spectral problems

Johannes Sjöstrand, Maciej Zworski (2007)

Annales de l’institut Fourier

We describe a simple linear algebra idea which has been used in different branches of mathematics such as bifurcation theory, partial differential equations and numerical analysis. Under the name of the Schur complement method it is one of the standard tools of applied linear algebra. In PDE and spectral analysis it is sometimes called the Grushin problem method, and here we concentrate on its uses in the study of infinite dimensional problems, coming from partial differential operators of mathematical...

Energy Critical nonlinear Schrödinger equations in the presence of periodic geodesics

Sebastian Herr (2010)

Journées Équations aux dérivées partielles

This is a report on recent progress concerning the global well-posedness problem for energy-critical nonlinear Schrödinger equations posed on specific Riemannian manifolds M with small initial data in H 1 ( M ) . The results include small data GWP for the quintic NLS in the case of the 3 d flat rational torus M = 𝕋 3 and small data GWP for the corresponding cubic NLS in the cases M = ℝ 2 × 𝕋 2 and M = ℝ 3 × 𝕋 . The main ingredients are bi-linear and tri-linear refinements of Strichartz estimates which obey the critical scaling, as well...

Envelopes of holomorphy for solutions of the Laplace and Dirac equations

Martin Kolář (1991)

Commentationes Mathematicae Universitatis Carolinae

Analytic continuation and domains of holomorphy for solution to the complex Laplace and Dirac equations in 𝐂 n are studied. First, geometric description of envelopes of holomorphy over domains in 𝐄 n is given. In more general case, solutions can be continued by integral formulas using values on a real n - 1 dimensional cycle in 𝐂 n . Sufficient conditions for this being possible are formulated.

Équation anisotrope de Navier-Stokes dans des espaces critiques.

Marius Paicu (2005)

Revista Matemática Iberoamericana

We study the tridimensional Navier-Stokes equation when the value of the vertical viscosity is zero, in a critical space (invariant by the scaling). We shall prove local in time existence of the solution, respectively global in time when the initial data is small compared with the horizontal viscosity.

Équations de champ moyen pour la dynamique quantique d’un grand nombre de particules

Patrick Gérard (2003/2004)

Séminaire Bourbaki

L’objet de cet exposé est de montrer comment l’évolution de Schrödinger pour le problème à N corps quantique est approchée, lorsque N tend vers l’infini, dans un régime convenable, par une évolution non-linéaire en dimension trois d’espace. On traitera le cas des bosons, qui conduit à l’équation de Schrödinger-Poisson, et celui des fermions, qui débouche sur le système de Hartree-Fock.

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