Finite element heterogeneous multiscale method for transient wave propagation

A finite element heterogeneous multiscale method (FE-HMM) is proposed for the time dependent wave equation with highly oscillatory, albeit not necessarily periodic, coefficients. It is based on a finite element discretization of an effective wave equation at the macro scale, whose a-priori unknown effective coefficients are computed “on the fly” on sampling domains within each macro finite element at the micro scale ε < 0. Since the sampling domains scale in size with ε, which corresponds to the finest scales in the possibly highly heterogeneous medium, the computational work is independent of ε. In [1], we proved optimal error estimates in the energy norm and the L2-norm with respect to the micro and macro scale mesh parameters, h and H, and also convergence to the homogenized solution as ε → 0.