An iterative method for elliptic problems with rapidly oscillating coefficients

Abstract

We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address different length scales. However, we use here the homogenized equation on all scales larger than a fixed multiple of the scale of oscillation of the coefficients. While the performance of standard multigrid methods degrades rapidly under the regime of large scale separation that we consider here, we show an explicit estimate on the contraction factor of our method which is independent of the size of the domain. We also present numerical experiments which confirm the effectiveness of the method, with openly available source code.

Optimal Artificial Boundary Condition for Random Elliptic Media

“…Representative volume method, a popular approach used by engineers, are systematically analyzed in [Fis19,KKO20]. Iterative multigrid methods have been studied in [Mou19,HMS21,AHKM21]. There have also been abundant research in numerical homogenization where the coefficients do not necessarily arise from random setting.…”