Numerical model reduction with error control in computational homogenization of transient heat flow

Abstract

Numerical Model Reduction (NMR) is exploited for solving the finite element problem on a Representative Volume Element (RVE) that arises from the computational homogenization of a model problem of transient heat flow. Since the problem is linear, an orthogonal basis is obtained via the classical method of spectral decomposition. A symmetrized version of the space–time variational format is adopted for estimating the error from the model reduction in (i) energy norm and in (ii) given Quantities of Interest. This technique, which was recently developed in the context of the (non-selfadjoint) stationary diffusion–convection problem, is novel in the present context of NMR. By considering the discrete, unreduced, model as exact, we are able to obtain guaranteed bounds on the error while using only the reduced basis and with minor computational effort. The performance of the error estimates is demonstrated via numerical results, where the subscale is modeled in both one and three spatial dimensions.