This article discusses the adaptation and convergence analysis by addressing an evolution equation employing composite finite element method (abbreviation CFE), incorporating one memory term, which is of positive type. The proposed method is one modification of the standard finite element method (FEM). The benefit of employing this technique is that it uses two-scale discretization of the domain-one with coarse-scale grid having mesh size H, another fine-scale grid having mesh size h. This helps in reducing the dimension of the considered domain space. In this article, an integro-differential equation of Volterra type is considered and we analyze the convergence using the proposed method, using the provided initial condition and boundary condition. The memory term is a convolution product of an elliptic operator and the positive-definite kernel. We discuss the fully discrete estimation for the error analysis, where both the time and space coordinates gets discretized. It is noted that the results obtained are optimal. We have carried out the numerical experiments for the validation of the conceptual outcomes.