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.

翻译：我们建立了若干常用有限元（FE）非线性全离散解的存在性理论以及线性化迭代的收敛性理论。首先，分别针对标准有限元、SUPG和边平均方法，证明了在特定条件下刚度矩阵为列M-矩阵，进而利用Brouwer不动点定理给出了这三种有限元非线性全离散解的存在性理论。其次，证明了常用线性化迭代方法——Gummel迭代的压缩性，并建立了该迭代的收敛性理论。最后，通过数值实验验证了上述理论。