The augmented drift-diffusion current equation, which includes velocity overshoot effects through the space derivatives of the electric field, cannot be directly extended beyond one dimension. A new formalism is developed which considers the carrier heating and the distribution relaxation effects to obtain a multidimensional augmented drift diffusion current equation. The equivalent mobility containing the velocity overshoot correction is derived from the perturbation analysis on the carrier temperature using the energy balance equation. The issues related to the numerical implementation of this generalized model and the validity of the assumptions are also discussed. >