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In this work we propose an efficient black-box solver for two-dimensionalstationary diffusion equations, which is based on a new robust discretizationscheme. The idea is to formulate an equation in a certain form withoutderivatives with a non-local stencil, which leads us to a linear system ofequations with dense matrix. This matrix and a right-hand side are representedin a low-rank parametric representation -- the quantized tensor train (QTT-)format, and then all operations are performed with logarithmic complexity andmemory consumption. Hence very fine grids can be used, and very accuratesolutions with extremely high spatial resolution can be obtained. Numericalexperiments show that this formulation gives accurate results and can be usedup to $2^{60}$ grid points with no problems with conditioning, while totalcomputational time is around several seconds.

Robust discretization in quantized tensor train format for elliptic problems in two dimensions