An adaptive Chebyshev iterative method

For the numerical solution of a boundary-value problem of three-dimensional elliptic equations an adaptive Chebyshev iterative method is constructed. In this adaptive method, the unknown lower bound of the spectrum of the discrete operator is refined in the additional cycle of the iterative method; the upper bound of the spectrum is taken to be its estimate by the Gershgorin theorem. Such procedure ensures the convergence of the constructed adaptive method with computational costs close to the costs of the Chebyshev method, which uses the exact boundaries of the spectrum of the discrete operator.