Multigrid method for periodic heterogeneous media Part 1: Convergence studies for one-dimensional case Original Research Article

A multi-grid method for a periodic heterogeneous medium in 1-D is presented. Based on the homogenization theory, special integrid transfer operators have been developed to simulate a low frequency response of the differential equations with oscillatory coefficients. The proposed multi-grid method has been proved to have a fast rate of convergence governed by the ratio q(4−q), where 0 < q ⩽ 1 depends on the microstructure. This estimate reveals that the rate of convergence increases as q 0, which corresponds to the increasing material heterogeneity. Numerical results have been found to be in good agreement with the theoretical estimate.