Superfast method with guaranteed accuracy for elliptic equations in rectangular domain

In finite-difference solution of elliptic equations we face with algebraic systems of enormous sizes with strongly rarefied matrices. A superfast iterative technique has been proposed that is viable for a wide class of problems. The method is based on the relaxation count for economic evolutionary factorized scheme using special set of steps constructed in logarithmic scale. The iterations convergence is proved to be exponential. The superfast convergence rate makes it possible to solve elliptic equations on multiply densening spatial grids with Richardson extrapolation applied. The latter provides a posteriori asymptotically precise error estimations for the grid solution.