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In this paper, we study a weak Galerkin (WG) finite element method for semiconductor device simulations. We consider the one-dimensional drift–diffusion (DD) and high-field (HF) models, which involves not only first derivative convection terms but also second derivative diffusion terms, as well as a coupled Poisson potential equation. The main difficulties in the analysis include the treatment of the nonlinearity and coupling of the models. The weak Galerkin finite element method adopts piecewise polynomials of degree k for the approximations of electron concentration and electric potential in the interior of elements, and piecewise polynomials of degree k + 1 for the discrete weak derivative space. The optimal order error estimates in a discrete H 1 norm and the standard L 2 norm are derived. Numerical experiments are presented to illustrate our theoretical analysis. Moreover, numerical schemes also work out for the discontinuous diffusion coefficient problems. [ABSTRACT FROM AUTHOR]

We continue our effort in Li et al. (2024) to develop a stabilizer-free weak Galerkin finite element method (SFWG-FEM) for the two-dimensional unsteady-state drift-diffusion (DD) model. The transient state equation includes both the first derivative convection and the second derivative diffusion terms, and it couples with a Poisson potential equation. The piecewise P k (k ≥ 1) polynomials elements in the interior and on the boundaries are utilized for both electron concentration and electric potential functions. The vector function of [ P j (K) ] 2 (j ≥ j 0) for the discrete weak gradient space is used on each element K. The main difficulty is the treatment of the non-linearity and coupling of the system. Based on an introduced Ritz projection and two L 2 projection operators, optimal error estimates of O (Δ t + h k + 1) in the L 2 norm and O (Δ t + h k) in the energy-like norm are derived. Numerical experiments are presented to illustrate the theoretical analysis. • Optimal estimate of stabilizer-free WG method for drift-diffusion model is derived. • The method is flexible while reducing complexity in analyzing and programming. • A Ritz projection that can be applied to estimate other problems is introduced. [ABSTRACT FROM AUTHOR]