A new numerical strategy for the drift-diffusion equations based on bridging the hybrid mixed and exponential fitted methods

We present a new discretization scheme to solve the stationary drift-diffusion equations based on the hybrid mixed finite element method. A convenient change of variables is adopted and the partial differential equations of the system are decoupled and linearized through Gummel's map. This gives rise to three equations that need to be solved in a staggered fashion: one of reaction-diffusion type (Poisson) and two exhibiting a diffusion-reaction character (continuity equations). The Poisson's equation is solved by the classical hybrid mixed finite element method, while the continuity equations are discretized by a new version of the hybrid mixed exponential fitted method. The novelty here lies on the bridging terms between Poisson and each continuity equation, pursued by exploring direct relations between the Lagrange multipliers, thereby avoiding the use of a projection operator. The static condensation technique is adopted to reduce the number of degrees of freedom. Moreover, the finite dimensional functional spaces characterizing the hybrid mixed methods are chosen to ensure that the discrete problems satisfy the discrete maximum principle when a mesh of rectangular elements is used. Numerical experiments simulating semiconductor devices are presented, showing that the proposed methodology is capable of producing solutions free from spurious oscillations and accurate fluxes without the need of highly refined or complex meshes.