In this paper we develop a residual based a posteriori error analysis for an augmentedmixed finite element method applied to the problem of linear elasticity in the plane.More precisely, we derive a reliable and efficient a posteriori error estimator for thecase of pure Dirichlet boundary conditions. In addition, several numericalexperiments confirming the theoretical properties of the estimator, andillustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution, are also reported.

In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of thelower Bloch spectrum. For the three dimensional linear elasticity system,the first eigenvalue is degenerate of multiplicity three and henceexistence of such a regular Bloch spectrum is not guaranteed. Theaim here is to develop all necessary spectral tools to overcome thesedifficulties. The existence of a directionally regular Bloch spectrum isproved and isused in the homogenization. As a consequence an interesting relation betweenhomogenization process and wave propagation in the homogenized medium isobtained. Existence of a spectral gap for the directionally regular Bloch spectrum is established and as a consequenceit is proved that higher modes apart from the first three do not contribute to the homogenization process.