This thesis carries out some studies concerning the mathematical modelling and the numerical approximation of problems involving homogenized metamaterials. In the first part, we investigate wave propagation problems with homogenized metamaterials for Maxwell's equations and acoustics or linear elasticity systems. We establish that each of these systems is well-posed under assumptions that are relevant for some models already designed in the literature. We next tackle their numerical approximation. We give results showing that the finite element method for the approximation of Helmholtz equation, when metatmaterials are involved, may not converges. We propose then a numerical scheme, the EF-AL scheme, which can be used with metamaterials and we prove that it converges as soon as the considered problem is well-posed. We finish studying the discontinuous galerkin scheme. We show numerically its convergence for some examples of metamaterials. The second part presents a formal non-periodic homogenization process for acoustic metamaterials. The work of A.G. Ramm, designing a medium by assembling many obstacles, is extended meanwhile accurately describing the asymptotic behavior of the field scattered by a finite number of small ball of radius \delta. The method of matched asymptotic expansions is used. We establish existence and uniqueness of the latter expansion and prove error estimates giving it a theoretical background. Next we assume that the number of small obstacles grows to infinity when \delta goes to 0 and pass to the limit in the asymptotic expansion. A Born approximation then yields the refractive index of an effective medium packaging all the small bodies which corresponds, in some cases, to a metamaterial.
Mathematical study and numerical approximation of homogenized metamaterials