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This paper aims to study the Galerkin finite element method (FEM) of both the semi-discrete and the fully-discrete schemes with linear triangular element for the time-dependent Joule heating problem. Through some new estimating approaches such as the different mathematical induction assumption, the mean-value technique and the special characters of this element, the superclose and superconvergence estimates for the related variables in the H 1 -norm are derived under lower regularities of the solutions of the considered problem rigorously, which simply the proof process and improve the corresponding results in the existing literature. Finally, numerical results are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, the discontinuous Galerkin method (DGM) of nonconforming Wilson element is studied for the semi-linear parabolic problem. The global superconvergence with respect to the mesh size are derived in the modified H 1 -norm for the semi-discrete scheme and two fully discrete schemes, in which the usual extrapolation and interpolation post-processing approaches are not involved, and the error estimates are one order higher than that of the traditional Galerkin finite element method (FEM). Therefore, the corresponding results in the existing literature are improved. Finally, some numerical results are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]