Discontinuous Galerkin finite element method for parabolic problems

In this paper, we develop a time and its corresponding spatial discretization scheme, based upon the assumption of a certain weak singularity of ‖ut(t)‖L2(Ω)=‖ut‖2‖ut(t)‖L2(Ω)=‖ut‖2, for the discontinuous Galerkin finite element method for one-dimensional parabolic problems. Optimal convergence rates in both time and spatial variables are obtained. A discussion of automatic time-step control method is also included.