Analysis of a heterogeneous multiscale fem for problems in elasticity

TLDR: In this paper, a finite element method (FEM) is used to discretize the physical problem of linear elasticity in a macroscopic FEM coupled with a microscopic FEM resolving the micro scale on small cells or patches.

Abstract: This paper is concerned with a finite element method (FEM) for multiscale problems in linear elasticity. We propose a method which discretizes the physical problem directly by a macroscopic FEM, coupled with a microscopic FEM resolving the micro scale on small cells or patches. The assembly process of the unknown macroscopic model is done without iterative cycles. The method allows to recover the macroscopic properties of the material in an efficient and cheap way. The microscale behavior can be reconstructed from the known micro and macro solutions. We give a fully discrete convergence analysis for the proposed method which takes into account the discretization errors at both micro and macro levels. In the case of a periodic elastic tensor, we give a priori error estimates for the displacement and for the macro and micro strains and stresses as well as an error estimate for the numerical homogenized tensor.