Advances in error estimation for homogenisation

In this paper, the concept of modeling error is extended to the homogenisation of elliptic PDEs. The main difficulty is the lack of a full description of the diffusion coefficients. We overcome this obstacle by representing them as a random a field. Under this framework, it is possible to quantify the accuracy of the surrogate model (the homogenised model) in terms of first moments of the energy norm and quantities of interest. This work builds on the seminal work of [1]. The methodology here presented rely on the Constitutive Relation Error (CRE) which states
that a certain measures of the primal and dual surrogate model upper bound the exact error. The surrogate model, in agreement with homogenisation, is deterministic. This property exploited to obtain bounds whose computation is also deterministic. It is also shown that minimising the CRE in the set of homogenisation schemes leads us to an optimal surrogate that is closely related to the classical Voigt and Reuss models. Numerical examples demonstrate that the bounds are easy and affordable to compute, and useful as long as the mismatch between he diffusion coefficients of the microstructure remain small. In the case of high mismatch, extensions are proposed, through the introduction of stochastic surrogate models.. [1]Romkes, Albert, J. Tinsley Oden, and Kumar Vemaganti."Multi-scale goal-oriented adaptive modeling of random heterogeneous materials." Mechanics of materials 38.8(2006): 859-872.