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We consider the laminar viscous channel ow over a porous surface. The size of the pores is much smaller than the size of the channel and it is important to determine the e ective boundary conditions at the porous surface. After studying the corresponding boundary layers, we obtain by a rigorous asymptotic expansion Sa man's modi cation of the interface condition observed by Beavers and Joseph. The e ective coe cient in the law is determined through an auxiliary boundary-layer type problem, whose computational and modeling aspects are discussed in detail. Furthermore, the approximation errors for the velocity and for the e ective mass ow are given as powers of the characteristic pore size ". Finally, the interface condition linking the e ective pressure elds, in the porous medium and in the channel, is given and the jump of the e ective pressures is explicitly determined.

In this paper, we develop multiscale methods appropriate for the homogenization of processes in domains containing thin heterogeneous layers. Our model problem consists of a nonlinear reaction-diffusion system defined in such a domain, and properly scaled in the layer region. Both the period of the heterogeneities and the thickness of the layer are of order ε. By performing an asymptotic analysis with respect to the scale parameter ε we derive an effective model which consists of the reaction-diffusion equations on two domains separated by an interface together with appropriate transmission conditions across this interface. These conditions are determined by solving local problems on the standard periodicity cell in the layer. Our asymptotic analysis is based on weak and strong two-scale convergence results for sequences of functions defined on thin heterogeneous layers. For the derivation of the transmission conditions, we develop a new method based on test functions of boundary layer type.

The formation of plaques is one of the main causes for the blockage of arteries. This can lead to ischaemic brain or myocardial infarctions as well as other cardiovascular diseases. Possible biochemical and biomechanical processes contribute to the development of plaque growth and rupture. The main biochemical processes are the penetration of monocytes and the accumulation of foam cells in the vessel wall, leading to the formation and growth of plaques. The biomechanical forces can be measured by observing stresses in the blood flow and the vessel wall, which may lead to the rupture of plaques.In this thesis, we formulate an appropriate model to describe the evolution of plaques. The model consists of both the interaction between the blood flow and the vessel wall, and the growth of plaques due to the penetration of monocytes from the blood flow into the vessel wall. The Navier-Stokes equations and the elastic structure equations are used to describe the dynamics of fluid (blood flow) and the mechanics of structure (vessel wall). The motion of monocytes is described by the convection-diffusion-reaction equation, coupled with an equation for the accumulation of foam cells. Finally the metric of growth is introduced to accurately determine the stress tensor, and its evolution equation is derived. The variational formulation of the model is transformed into the ALE (Arbitrary Lagrangian-Eulerian) formulation, and all the equations are rewritten in the fixed domain. Temporal discretization is achieved with finite differences and spatial discretization is based on the Galerkin finite element method. The nonlinear systems are linearized and solved by the Newton method.Based on the model and the numerical methods above, numerical simulations are performed by using the software Gascoigne. The obtained numerical results make an agreement with the observation, and support the assumption that the penetration of monocytes and the accumulation of foam cells lead to the formation and growth of plaques, and that the evolution of plaques induces the increase of stresses in the vessel wall, which is an indicator of plaque rupture. Zusammenfassung

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In this article, we derive approximations and effective boundary laws for solutions u " of the Poisson equation on a domain " & R n whose boundary differs from the smooth boundary of a domain & R n by rapid oscillations of size ". We construct a boundary layer correction which yields an O(") approximation in the energy norm and an Oð" 3=2 Þ approximation in the L 2 -norm for a right-hand side f 2 L 2 ð [ " Þ. We also show that the same approximation order can be obtained for the error on subdomains by solving an effective equation on satisfying a boundary condition of Robin type.

A Galerkin approach for a class of multiscale reaction-diffusion systems with nonlinear coupling between the microscopic and macroscopic variables is presented. This type of models are obtained e.g. by upscaling of processes in chemical engineering (particularly in catalysis), biochemistry, or geochemistry. Exploiting the special structure of the models, the functions spaces used for the approximation of the solution are chosen as tensor products of spaces on the macroscopic domain and on the standard cell associated to the microstructure. Uniform estimates for the finite dimensional approximations are proven. Based on these estimates, the convergence of the approximating sequence is shown. This approach can be used as a basis for the numerical computation of the solution.