An Adaptive Chebyshev Iterative Method

Abstract

An adaptive Chebyshev iterative method used to solve boundary-value problems for three-dimensional elliptic equations numerically is constructed. In this adaptive method, the unknown lower bound of the spectrum of the discrete operator is refined in the additional iteration cycle, and the upper bound of the spectrum is taken to be its estimate by the Gershgorin theorem. Such a procedure ensures the convergence of the constructed adaptive method with the computational costs close to the costs of the standard Chebyshev method, which uses the exact bounds of the spectrum of the discrete operator.