A linear triangular finite element method-implicit difference scheme (FEM-IDS) for the solution of second order strongly damped wave equation (SDWE) with memory on domain with interfaces is proposed. Sufficient conditions that guarantee the existence of a unique solution are given. The finite element discretization is such that the arbitrary (but smooth) interface is first approximated by a polygon with a boundary whose vertices all lie on the interface. With the interface being at σ-distance from the approximate interface, linear interpolation operator incorporating the influence of σ isproved. This together with Ritz–Volterra operator and some auxiliary error estimates in the neighborhood of the interface are used to obtain the convergence estimate of the proposed scheme. The scheme is applied to some test examples and the numerical results confirm the computational efficiency of the method.