Sparse tensor finite elements for elastic wave equation with multiple scales

We consider an elastic wave equation whose moduli depend on nn separable microscopic scales and are periodic with respect to each scale in a domain DD in RdRd. By using multiscale convergence, we deduce an (n+1)d(n+1)d dimensional problem that contains all the macroscopic and microscopic information on the multiscale equation. The full tensor product finite element approach is natural to solve this high dimensional multiscale homogenized equation but is too expensive practically. We develop a sparse tensor finite element approach that achieves essentially equal accuracy as the full tensor product finite elements but requires an essentially equal number of degrees of freedom as for solving a problem in RdRd. Restricting our consideration to the case of zero initial wave displacement, we deduce a corrector with a homogenization error estimate in the case of two scales. From this numerical correctors are constructed from the finite element solutions of the high dimensional multiscale homogenized problem with an error of the order of the sum of the homogenization and the finite element errors. For the general multiscale problems, we construct numerical correctors from the finite element solutions but without an explicit error. Numerical examples on two and three scale elastic wave equations in two dimensions confirm our analysis.