Multiscale Asymptotic Expansion And Finite Element Methods For Periodic Composite Materials Hyperbolic Type Wave Problems

A class of important mathematical physics problems, i.e. hyperbolic type wave problems are discussed in this paper. First, we use homogenization theory and multiscale asymptotic expansion to resolve hyperbolic type wave problems with rapidly oscillating coefficients in periodic composite materials. We capture a more easily operated asymptotic expansion when computed in practice under the assumption that the oscillating coefficients is of two scales and is periodic in the fast scale, and provide a detailed convergence analysis of our method. Second, we use multiscale finite element method to discuss the approximation of semi-discrete resolution about the hyperbolic type wave equation with rapidly oscillating in small periodic composite materials, and provide its error estimates.