Gridding and Discretization For Divergence Form (Semiconductor-Like) PDEs
We develop the box method for solving partial differential equations in the divergence (or conservation law) form. We first use graphic theoretical approaches to generate grids, i.e. partition the governing domain into boxes. Then we apply Green’s theorem box-wisely and the constant-j assumption edge-pair-wisely to obtain the discretization system of equations, which lead to the numerical solution to the problem. We show that the box method is inherently more efficient than the traditional finite element method for linear (or convection-diffusion) problems in and 2 dimensions. We also present some potential thoughts on implementing the box method for problems in higher dimensions and with nonlinearity.