A method of computing the error in the solution of the semiconductor current continuity equations as well as the error in the terminal currents is proposed. An appropriate error indicator is developed based on a divergence free upwinding (finite element) discretization. The grid used for discretization is adapted to the error in the solution by dynamically adding or removing grid points in order to improve the solution and thus the terminal currents. The examples indicate that it is sufficient only for a reverse-biased p-n junction to refine the grid according to the error in the Poisson equation. In the forward biased case, it is necessary to take into account the error in the current continuity equation in order to guarantee exact terminal currents.>

A hybrid finite-element method to discretize the continuity equation in semiconductor device simulation is given. Within each element of a finite element discretization, the current is uniquely determined by nodal values of the density and the potential. The authors use the integrability condition for a system of partial differential equations to obtain the equations that determine the current within the element. They then satisfy the continuity in the current flow across interelement boundaries in a weak sense. They have found that the method works in any dimension and for (d-dimensional) simplexes as well as for quadrilaterals, bricks, prisms, and so on, although they have no proof that it will not break down in particular cases.>