Abstract
We present an efficient method for the computation of homogenized coefficients of divergence-form operators with random coefficients. The approach is based on a multiscale representation of the homogenized coefficients. We then implement the method numerically using a finite-element method with hierarchical hybrid grids, which is a semi-implicit method allowing for significant gains in memory usage and execution time. Finally, we demonstrate the efficiency of our approach on two- and three-dimensional examples, for piecewise-constant coefficients with corner discontinuities. For moderate ellipticity contrast and for a precision of a few percentage points, our method allows to compute the homogenized coefficients on a laptop computer in a few seconds, in two dimensions, or in a few minutes, in three dimensions.
Highlights
We consider operators of the form ∇ · a∇, where a = (a(x))x∈Rd is a random coefficient field on Rd taking values in the set of symmetric positive definite matrices. We assume that this random coefficient field is uniformly elliptic, Zd-stationary, and of unit range of dependence; see Section 2.1 for precise statements
We explain how to implement the algorithm in practice, using the notion of hierarchical hybrid grids, and demonstrate its performance on examples
We introduce notions related to stationary random fields and solutions, and recall the definition of the homogenized coefficients in terms of correctors
Summary
The goal of this paper is to define, study, and implement an efficient approach to the calculation of homogenized coefficients for divergence-form operators with random coefficients. Compared with the discrete setting of finite-difference operators investigated in [52], the case of continuous differential operators considered here poses crucial new challenges from a computational perspective With applications such as those in materials science in mind, it is natural to consider piecewise constant coefficient fields. The algorithm as proposed here would run just fine, but it would compute the homogenized matrix associated with the particular finite-element discretization that is chosen; if the discretization is coarse, this homogenized matrix will be far from the homogenized matrix of the continuous operator To remedy this problem, we need to rely on much finer discretizations of the coefficient field. The term nd 2 in (1.7) will have to be suitably modified. (This makes the development of a more adaptive algorithm appealing, since such an algorithm could automatically select the optimal scalings without supervision.) in view of [8, 23], we expect that the results can be generalized to the case of perforated domains of percolation type
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
