Application of the Composite Finite Element Framework for Evolution Equation with Nonsmooth Kernel in Smooth Domain

This research paper mainly focuses on the adaptation and convergence analysis of the Composite Finite Element (CFE) method for the evolution equation with a positive-type memory term. The CFE method is one variant of the standard finite element method. The advantage of the method is that it uses two-scale discretization for the domain-one with coarse-scale grid with mesh size [Formula: see text] and another fine-scale grid with mesh size [Formula: see text]. This helps in reducing the dimension of the considered domain space. In this paper, we consider an initial-boundary value problem for an integro-differential equation of the Volterra-type and we study the convergence using the proposed method. 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 theoretical results.