Abstract
In this paper, we develop a time and its corresponding spatial discretization scheme, based upon the assumption of a certain weak singularity of ‖ u t ( t ) ‖ L 2 ( Ω ) = ‖ u t ‖ 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.
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