Fully discrete analysis of the heterogeneous multiscale method for elliptic problems with multiple scales

A fully discrete analysis of the finite element heterogeneous multiscale method (FE-HMM) for elliptic problems with N+1 well-separated scales is discussed. The FE-HMM is a numerical homogenization method that relies on a macroscopic scheme (macro FEM) for the approximation of the effective solution corresponding to the multiscale problem. The effective data are recovered from micro scale computation (micro FEM) on sampling domains located at appropriate quadrature points of the macroscopic mesh. At the macroscopic level, the numerical method can be seen as an FEM with numerical quadrature for a modified effective problem, hence variational crimes are made when designing this method. Up to now, the method has been analysed for two scales and the micro FEM was assumed to be conforming. For more than two scales, variational crimes are committed also at the intermediate (meso) scales and the effective data of the macroscopic scheme are obtained from a cascade of FEMs with numerical integration, which require a careful analysis. Numerical experiments for three-scale problems illustrate the theoretical convergence rates.