Finite element heterogeneous multiscale method for elastic waves in heterogeneous media

Abstract

A finite element heterogeneous multiscale method (FE-HMM) is proposed for the simulation of time-dependent elastic waves in a rapidly varying heterogeneous elastic medium. It is based on a standard finite element discretization of an effective wave equation at the macro scale, whose a priori unknown effective material coefficients are computed on sampling domains at the micro scale within each macro finite element. Hence the computational effort becomes independent of the highly heterogeneous elastic medium at the smallest scale. Optimal error estimates and convergence rates in the energy and the L2 norm are derived, which are explicit in the macro and micro discretization errors. Numerical experiments verify the sharpness of the error bounds and illustrate the versatility of the method for non-periodic, layered or stochastic media.