  The goal of this paper is to develop a numerical algorithm that solves a
two-dimensional elliptic partial differential equation in a polygonal domain
using tensor methods and ideas from isogeometric analysis. The proposed
algorithm is based on the Finite Element (FE) approximation with Quantized
Tensor Train decomposition (QTT) used for matrix representation and solution
approximation. In this paper we propose a special discretisation scheme that
allows to construct the global stiffness matrix in the QTT-format. The
algorithm has $O(\log n)$ complexity, where $n=2^d$ is the number of nodes per
quadrangle side. A new operation called z-kron is introduced for QTT-format. It
makes it possible to build a matrix in z-order if the matrix can be expressed
in terms of Kronecker products and sums. An algorithm for building a QTT
coefficient matrix for FEM in z-order "on the fly", as opposed to the
transformation of a calculated matrix into QTT, is presented. This algorithm
has $O(\log n)$ complexity for $n$ as above.
