High dimensional finite elements for multiscale wave equations

Abstract

For locally periodic multiscale wave equations in $\mathbb{R}^d$ that depend on a macroscopic scale and n microscopic separated scales, we solve the high dimensional limiting multiscale homogenized problem that is posed in $(n+1)d$ dimensions and is obtained by multiscale convergence. We consider the full and sparse tensor product finite element methods, and analyze both the spatial semidiscrete and the fully (both temporal and spatial) discrete approximating problems. With sufficient regularity, the sparse tensor product approximation achieves a convergence rate essentially equal to that for the full tensor product approximation, but requires only an essentially equal number of degrees of freedom as for solving an equation in $\mathbb{R}^d$ for the same level of accuracy. For the initial condition $u(0,x)=0$, we construct a numerical corrector from the finite element solution. In the case of two scales, we derive an explicit homogenization error which, together with the finite element error, produces an explicit rate of convergence for the numerical corrector. Numerical examples for two- and three-scale problems in one or two dimensions confirm our analysis.