MiS Preprint Repository

We study separability properties of solutions of elliptic equations with piecewise constant coefficients in ${\mathbb{R}}^d$, $d\ge 2$. Besides that, we develop efficient tensor-structured preconditioner for the diffusion equation with variable coefficients. It is based only on rank structured decomposition of the tensor of reciprocal coefficient and on the decomposition of the inverse of the Laplacian operator. It can be applied to full vector with linear-logarithmic complexity in the number of unknowns $N$. It also allows low-rank tensor representation, which has linear complexity in dimension $d$, hence, it gets rid of the "curse of dimensionality" and can be used for large values of $d$. Extensive numerical tests are presented.