In this work we construct an asymptotic expansion to the weak solution of anisotropic singular perturbation problems of elliptic type. The paper aims to go deep in the study of the rate of convergence. In fact the analysis of the asymptotic expansion strongly helps in understanding what is, and what is not, needed to improve the rate and the type of convergence. In order to define the coefficients of the development some smoothness basic properties (with respect to the parameters) of elliptic problems are established.
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