Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

Abstract

We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned conjugate gradient iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver, we numerically investigate the convergence of the representative volume element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in Gloria et al. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.

An Efficient Online-Offline Method for Elliptic Homogenization Problems

“…Secondly, for x = A,B,C, we measure the variance as , which is consistent with the theoretical predictions [25]. A systematical numerical tests for the variance may be found in a recent work [31].…”

Computing homogenized coefficientsviamultiscale representation and hierarchical hybrid grids

“…Indeed, for any α > 0, solutions can develop singularities that fail to be in H 1+α , provided that the ellipticity contrast is sufficiently large, and standard finite-element methods provide approximations of these singular solutions that converge in L 2 at a rate that is bounded below by c h 1+α . In fact, for coefficient fields arranged in a checkerboard-type pattern in two dimensions, as considered in [43] and in the present paper, one can identify exactly the optimal exponent of regularity in terms of the ellipticity contrast: solutions are H β -regular if and only if β < 1 + α, where α is given in (5.5), as proved in [57] and recalled in Subsection 5.1 below. In particular, an asymptotic rate of convergence in L 2 of O(h 3 2 ) can only be obtained for values of the ellipticity contrast Λ below 3 + 2 √ 2.…”

“…For coefficient fields that are very similar to those we investigate numerically here, the standard representative volume method was combined with a tensor-based discretization scheme in [43] to compute homogenized matrices, in two dimensions. The authors of [43] state that their numerical approximation method displays an empirical rate of convergence in L 2 of O(h β ) with β ⩾ 3 2, where h measures the size of a discretized element. We believe that this is an artefact of pre-asymptotic effects and moderate ellipticity contrast.…”

Tensor-based techniques for fast discretization and solution of 3D elliptic equations with random coefficients

“…The FDM discretization scheme on a tensor grid is described in detail, and the computational characteristics in terms of L for the 3D Matlab implementation of the PCG iteration are illustrated. The present work continues the developments in [22], where the numerical primer to study the asymptotic convergence rate vs. L for the homogenized matrix for 2D elliptic PDEs with random coefficients was investigated numerically. The presented elliptic problem solver can be applied for calculation of long sequences of stochastic realizations in numerical analysis of 3D stochastic homogenization problems for ergodic processes, for solving 3D quasi-periodic geometric homogenization problems, as well as in the numerical simulation of dynamical many body interaction processes and multi-particle electrostatics.…”