  We establish the existence theory of several commonly used finite element
(FE) nonlinear fully discrete solutions, and the convergence theory of a
linearized iteration. First, it is shown for standard FE, SUPG and
edge-averaged method respectively that the stiffness matrix is a column
M-matrix under certain conditions, and then the existence theory of these three
FE nonlinear fully discrete solutions is presented by using Brouwer's fixed
point theorem. Second, the contraction of a commonly used linearized iterative
method-Gummel iteration is proven, and then the convergence theory is
established for the iteration. At last, a numerical experiment is shown to
verifies the theories.
