  In this work we propose an efficient black-box solver for two-dimensional
stationary diffusion equations, which is based on a new robust discretization
scheme. The idea is to formulate an equation in a certain form without
derivatives with a non-local stencil, which leads us to a linear system of
equations with dense matrix. This matrix and a right-hand side are represented
in a low-rank parametric representation -- the quantized tensor train (QTT-)
format, and then all operations are performed with logarithmic complexity and
memory consumption. Hence very fine grids can be used, and very accurate
solutions with extremely high spatial resolution can be obtained. Numerical
experiments show that this formulation gives accurate results and can be used
up to $2^{60}$ grid points with no problems with conditioning, while total
computational time is around several seconds.
