A semi-implicit approach is proposed for computing the current density in superconductors characterized by nonlinear vectorial power law. A nodal discontinuous Galerkin method is adopted for the spatial discretization of the nonlinear system satisfied by the components of the electric field. Explicit developments are used to construct boundary conditions to avoid the modeling of a volume around the superconducting sample. A modified Newton iterative method is introduced for solving the discrete system. Numerical examples on a 2-D superconducting plate and a 3-D superconducting cube are computed. Distributions of a component of the current density are presented and differences in the diffusive process are highlighted. The penetration time and losses are compared with those obtained with an A-V formulation solved by a finite-volume method.