Finite Difference Scheme for Parabolic Problems on Composite Grids with Refinement in Time and Space

Finite difference schemes for transient convection-diffusion problems on grids with local refinement in time and space are constructed and studied. The construction utilizes a modified upwind approximation and linear interpolation at the slave nodes. The proposed schemes are implicit of backward Euler type and unconditionally stable. Error analysis is presented in the maximum norm, and convergence estimates are derived for smooth solutions. Optimal approximation results for ratios between the spatial and time discretization parameters away from the CFL condition are shown. Finally, numerical examples illustrating the theory are given.