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Analysis of patch substructuring methods

Martin Gander, Laurence Halpern, Frédéric Magoulès, Francois Roux (2007)

International Journal of Applied Mathematics and Computer Science

Patch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric patches reaching into neighboring subdomains condensated, on the interfaces to enhance the performance of the method, while keeping it non-overlapping. These methods are very convenient to use in practice, but their convergence properties have not been studied yet. We analyze geometric patch substructuring methods for the special case of one...

Anisotropic mesh adaption: application to computational fluid dynamics

Simona Perotto (2005)

Bollettino dell'Unione Matematica Italiana

In this communication we focus on goal-oriented anisotropic adaption techniques. Starting point has been the derivation of suitable anisotropic interpolation error estimates for piecewise linear finite elements, on triangular grids in 2 ⁢ D . Then we have merged these interpolation estimates with the dual-based a posteriori error analysis proposed by R. Rannacher and R. Becker. As examples of this general anisotropic a posteriori analysis, elliptic, advection-diffusion-reaction and the Stokes problems...

Approximation by generalized impedance boundary conditions of a transmission problem in acoustic scattering

Xavier Antoine, Hélène Barucq (2005)

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

This paper addresses some results on the development of an approximate method for computing the acoustic field scattered by a three-dimensional penetrable object immersed into an incompressible fluid. The basic idea of the method consists in using on-surface differential operators that locally reproduce the interior propagation phenomenon. This approach leads to integral equation formulations with a reduced computational cost compared to standard integral formulations coupling both the transmitted...

Approximation by generalized impedance boundary conditions of a transmission problem in acoustic scattering

Xavier Antoine, Hélène Barucq (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper addresses some results on the development of an approximate method for computing the acoustic field scattered by a three-dimensional penetrable object immersed into an incompressible fluid. The basic idea of the method consists in using on-surface differential operators that locally reproduce the interior propagation phenomenon. This approach leads to integral equation formulations with a reduced computational cost compared to standard integral formulations coupling both the transmitted...

Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity

Olivier Rey, Juncheng Wei (2005)

Journal of the European Mathematical Society

We show that the critical nonlinear elliptic Neumann problem Δ u − μ u + u 7 / 3 = 0 in Ω , u > 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω , where Ω is a bounded and smooth domain in ℝ 5 , has arbitrarily many solutions, provided that μ > 0 is small enough. More precisely, for any positive integer K , there exists μ K > 0 such that for 0 < μ < μ K , the above problem has a nontrivial solution which blows up at K interior points in Ω , as μ → 0 . The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

Francesca De Marchis, Isabella Ianni, Filomena Pacella (2015)

Journal of the European Mathematical Society

We consider the semilinear Lane–Emden problem where p > 1 and Ω is a smooth bounded domain of ℝ 2 . The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ( p ) , as p → + ∞ . Among other results we show, under some symmetry assumptions on Ω , that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as p → + ∞ , and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville...

Asymptotic and numerical modelling of flows in fractured porous media

Philippe Angot, Franck Boyer, Florence Hubert (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

This study concerns some asymptotic models used to compute the flow outside and inside fractures in a bidimensional porous medium. The flow is governed by the Darcy law both in the fractures and in the porous matrix with large discontinuities in the permeability tensor. These fractures are supposed to have a small thickness with respect to the macroscopic length scale, so that we can asymptotically reduce them to immersed polygonal fault interfaces and the model finally consists in a coupling between...

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