Kassem Mustapha, Hussein Mustapha; A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel. IMA J Numer Anal 2010; 30 (2): 555-578. doi: 10.1093/imanum/drn075
We study a generalized extrapolated Crank–Nicolson scheme for the time discretization of a semilinear integro-differential equation with a weakly singular kernel, in combination with a space discretization by linear finite elements. The scheme uses variable grids in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k2 + h2, where k and h are the parameters for the time and space meshes, respectively. The results of numerical computations demonstrate the convergence of our scheme.