Sparse tensor finite elements for elliptic multiple scale problems

Locally periodic, elliptic multiscale problems in a bounded Lipschitz domain $D\subset\mathbb{R}^n$ with $K\geq 2$ separated scales are reduced to an elliptic system of $K$ coupled, anisotropic elliptic one-scale problems in a cartesian product domain of total dimension $Kn$ (e.g.~\cite{N89,A,AB,CDG2008}). In \cite{HS,SSt08}, it has been shown how thesecoupled elliptic problems could be solved by sparse tensor wavelet Finite Element methods in log-linear complexity with respect to the number $N$ of degrees of freedom required by multilevel solvers for elliptic one-scale problems in $D$ with the same convergence rate.In the present paper, the high dimensional one-scale limiting problems are discretized by a sparse tensor product finite element method (FEM) with standard, one-scale FE basis functions as used in engineering FE codes. Sparse tensorizationand multilevel preconditioning is achieved by a BPX multilevel iteration. We show that the resulting sparse tensor multilevel FEM resolves all physical length scales throughout the domain, with efficiency (i.e.,~accuracy versus work and memory) comparable to that of multigrid solvers for elliptic one-scale problems in the physical domain $D$. In particular, our sparse tensor FEM gives numerical approximations of the correct homogenized limit as well as compressed numerical representationsof all first order correctors, throughout the physical domain with performance independent of the physical problem's scale parameters. Numerical examples with standard FE shape functions and BPX multilevelpreconditioners for elliptic problems with $K=2$ separated, physical scales in spatial dimension $n=2$ confirm the theoretical results. In particular, the present approach allows to avoid the construction of wavelet FE bases necessary in previous work \cite{HS,SSt08} while achieving resolution of all scales throughout the physical domain in log-linear complexity,with the logarithmic exponent behaving linearly in the number $K$ of scales.