Homogenization of the acoustic wave equation with a continuum of scales.

Abstract

We consider the acoustic wave equation in dimension n in situations where the bulk modulus and the density of the medium are only bounded. We show under a Cordes type condition that the second order derivatives of the pressure field with respect to harmonic coordinates are in L 2 (instead of H 1 with respect to Euclidean coordinates). It follows that it is possible to homogenize the wave equation without assumptions of scale separation or ergodicity by pre-computing n solutions of the associated elliptic equation (at a cost of O(N(log N) n+3 ) operations [19], where N stands for the number of degrees of freedom describing the fine scale structure of the medium).