High order finite element calculations for the deterministic Cahn-Hilliard equation

In this work, we propose a numerical method based on high degree continuous nodal elements for the Cahn-Hilliard evolution. The use of the p-version of the finite element method proves to be very efficient and favorably compares with other existing strategies (C^1 elements, adaptive mesh refinement, multigrid resolution, etc). Beyond the classical benchmarks, a numerical study has been carried out to investigate the influence of a polynomial approximation of the logarithmic free energy and the bifurcations near the first eigenvalue of the Laplace operator

A MULTISCALE CORRECTION METHOD FOR LOCAL SINGULAR PERTURBATIONS OF THE BOUNDARY

International audienceIn this work, we consider singular perturbations of the boundary of a smooth domain. We describe the asymptotic behavior of the solution uε of a second order elliptic equation posed in the perturbed domain with respect to the size parameter ε of the deformation. We are also interested in the variations of the energy functional. We propose a numerical method for the approximation of uε based on a multiscale superposition of the unperturbed solution u0 and a profile defined in a model domain. We conclude with numerical results

BEST DESIGN FOR A FASTEST CELLS SELECTING PROCESS

International audienceWe consider a cell sorting process based on negative dielectrophoresis. The goal is to optimize the shape of an electrode network to speed up the positioning. We first show that the best electric field to impose has to be radial in order to minimize the average time for a group of particles. We can get an explicit formula in the specific case of a uniform distribution of initial positions, through the resolution of the Abel integral equation. Next,we use a least-square numerical procedure to design the electrode's shape

Self-similar perturbation near a corner: matching versus multiscale expansions for a model problem

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Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer

Version April 9, 2004International audienceWe consider the solution of an interface problem posed in a bounded domain coated with a layer of thickness $\epsilon$ and with external boundary conditions of Dirichlet or Neumann type. Our aim is to build a multi-scale expansion as $\epsilon$ goes to $0$ for that solution. After presenting a complete multi-scale expansion in a smooth coated domain, we focus on the case of a corner domain. Singularities appear, obstructing the construction of the expansion terms in the same way as in the smooth case. In order to take these singularities into account, we construct profiles in an infinite coated sectorial domain. Combining expansions in the smooth case with splittings in regular and singular parts involving the profiles, we construct two families of multi-scale expansions for the solution in the coated domain with corner. We prove optimal estimates for the remainders of the multi-scale expansions

Computations of the first eigenpairs for the Schrödinger operator with magnetic field

International audienceThis paper is devoted to computations of eigenvalues and eigenvectors for the Schrödinger operator with constant magnetic field in a domain with corners, as the semi-classical parameter $h$ tends to $0$. The eigenvectors corresponding to the smallest eigenvalues concentrate in the corners: They have a two-scale structure, consisting of a corner layer at scale $\sqrt h$ and an oscillatory term at scale $h$. The high frequency oscillations make the numerical computations particularly delicate. We propose a high order finite element method to overcome this difficulty. Relying on such a discretization, we illustrate theoretical results on plane sectors, squares, and other straight or curved polygons. We conclude by discussing convergence issues

On generalized Ventcel's type boundary conditions for Laplace operator in a bounded domain

acceptée dans la revue SIAM J. Math. Anal.International audienceVentcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to wellposed variational problems under a sign condition of a coefficient. Nevertheless situations where this condition is violated appeared in several works. The wellposedness of such problems was still open. This manuscript establishes, in the generic case, the existence and uniqueness of the solution for the Ventcel boundary value problem without the sign condition. Then, we consider perforated geometries and give conditions to remove the genericity restriction

Artificial boundary conditions to compute correctors in linear elasticity

International audienceIn this paper, we derive artificial boundary conditions for the computation of correcting terms in a perturbed problem of linear elasticity. Theses conditions appear to be of Ventcel form, and lead to a non-coercive boundary value problem

Artificial conditions for the linear elasticity equations

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