A Crank-Nicolson Finite Element treatment of time-dependent singularities of the one dimensional Heat equation

Abstract The convergence rate of the Crank-Nicolson Finite Element Method (FEM) for the Heat equation can be affected if the solution entails time-dependent singularities. This paper presents a Crank-Nicolson FEM coupled to a Predictor-corrector algorithm to recover the optimal convergence rate when the solution has singularities. The FEM presented is based on the approximation of the time-dependent singular function by Fourier series. Numerical experiments are presented to show the efficiency of the method.