  In this paper, we consider the Poisson equation on a "long" domain which is
the Cartesian product of a one-dimensional long interval with a
(d-1)-dimensional domain. The right-hand side is assumed to have a rank-1
tensor structure. We will present and compare methods to construct
approximations of the solution which have tensor structure and the
computational effort is governed by only solving elliptic problems on
lower-dimensional domains. A zero-th order tensor approximation is derived by
using tools from asymptotic analysis (method 1). The resulting approximation is
an elementary tensor and, hence has a fixed error which turns out to be very
close to the best possible approximation of zero-th order. This approximation
can be used as a starting guess for the derivation of higher-order tensor
approximations by an alternating-least-squares (ALS) type method (method 2).
Numerical experiments show that the ALS is converging towards the exact
solution. Method 3 is based on the derivation of a tensor approximation via
exponential sums applied to discretised differential operators and their
inverses. It can be proved that this method converges exponentially with
respect to the tensor rank. We present numerical experiments which compare the
performance and sensitivity of these three methods.
