Abstract. Using tools from asymptotic analysis for elliptic equations, we study numerical methods in order to simulate the hemo-dynamics in a stented artery. Our aim is to derive accurate methods to discretize a multi-scale problems that are stable and accurate with respect to the very small size of the stent’s wires.

1. Introduction Atherosclerosis and rupture of aneurysm are lethal pathologies of the cardio-vascular system. A possible therapy consists in introducing a metallic multi-layered stent, either as a supplementary protection of the arterial wall or in order to slow vortices in the aneurysm and to favor coagulation of the sac. The stent is a complex multi-scale geometric structure that modifies locally the characteristics of the blood flow. In order to understand from the mathematical point of view the perturbation induced by the presence of a stent on the blood flow two possible strategies can be considered: • Either one uses homogenization theory and considers the stent as a fictitious interface [4, 1, 3]. In this case one obtains an averaged asymptotic expansion. This analysis provides error estimates: ku − u kL2 (Ω ) ≤ kO(α )

(1)

•

This • •

where is the (very small) diameter of a single wire of the stent, and u the exact solution for fixed while u is the averaged asymptotic expansion series. Or one uses a brute force strategy, nowadays possible thanks to the increasing computational power of the computers: by finite elements one discretizes the direct problem and provides u,h where h is now a discretisation parameter that should as be small. The accuracy of the numerical problem to be solved depends at this step on some higher derivative of the continuous solution targeted [2]:

ku − u,h kL2 Ω ≤ k Dλ u L2 (Ω ) , |λ| = k, k ≥ 2 √ but this continuous derivatives depend as well on and can behave as 1/ . master thesis should investigate the latter direction: to this aim, the candidate should derive numerical methods that are both stable and accurate when and h become small simultaneously, obtain discrete error estimates similar to (1) 2. prerequisites

The candidate should be interested by analysis and discretisation of elliptic partial differential equations (Laplace, Stokes, Naver-Stokes). In this field he should investigate in both aspects of multi-scale problems: • theory : homogenization of periodic structures • numerics : finite elements discretisation of elliptic problems References [1] D. Bresch and V. Milisic. High order multi-scale wall laws : part i, the periodic case. accepted for publication in Quart. Appl. Math. 2008. [2] A. Ern and J.-L. Guermond. Theory and Practice of Finite Elements, volume 159 of Applied Mathematical Series. Springer-Verlag, New York, 2004. [3] V. Miliˇsi´ c. Blood-flow modelling along and trough a braided multi-layer metallic stent. submitted. [4] E. S´ anchez-Palencia. Nonhomogeneous media and vibration theory, volume 127 of Lecture Notes in Physics. SpringerVerlag, Berlin, 1980.