| Author | Jacob Fish and Vladimir Belsky |
|---|---|
| Title | Adaptive Multi-Grid Method for a Periodic Heterogeneous Medium in 1-D |
| Year | 1994 |
| Pages | - - |
| Publisher | Springer Verlag |
| Editor | J. Flaherty, I. Babuska, W.D. Henshay, J.E. Oliger and P.A. Tezduyar |
| Abstract | A multi-grid method for a periodic heterogeneous medium in 1-D is presented. Based on the homogenization theory special intergrid connection operators have been developed to imitate 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 , where depends on the microstructure. This estimate reveals that the rate of convergence increases as , which corresponds to the increasing material heterogeneity. An adaptive multiscale computational scheme is developed. By this technique a computational model entirely constructed on the scale of material heterogeneity is only used where it is necessary to do so, or as indicated by so called Microscale Reduction Error (MRE) indicators, while in the remaining portion of the problem domain, the medium is treated as homogeneous with effective properties. Such a posteriori MRE indicators and estimators are developed on the basis of assessing the validity of two-scale asymptotic expansion. |