A Local-in-Space-Timestep Approach to a Finite Element Discretization of the Heat Equation with a Posteriori Estimates

A new numerical method is presented for the heat equation with discontinuous coefficients based on a Crank–Nicolson scheme and a conforming finite element space discretization. In the proposed method each node of the spatial discretization may have the global timestep split into an arbitrary number of local substeps in order to pursue a local improvement of the time discretization in the regions of the spatial domain where the solution changes rapidly. This method can possibly be used together with adaptive strategies for both the space discretization and the choice of timesteps to suitably produce an efficient space-time discretizaton of the problem. Robust a posteriori upper and lower bounds of the error are proposed to attain this target. Moreover, some indications are given on how to modify the mesh, the timestep, and the number of substeps to improve the discretization.