Monotone iterative methods for the adaptive finite element solution of semiconductor equations

• Published in: Journal of computational and applied mathematics Vol. 159; no. 2; pp. 341 - 364
• Main Authors: Chen, Ren-Chuen, Liu, Jinn-Liang
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.10.2003; Elsevier
• Subjects: Adaptive finite element; Drift-diffusion model; Exact sciences and technology; Mathematics; Monotone iteration; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Partial differential equations, boundary value problems; Sciences and techniques of general use; Adaptive finite element; Monotone iteration; Drift-diffusion model; Stiffness matrix; Semiconductor equation; Existence theorem; Jacobi method; Picard method; Gauss method; Iterative method; Numerical method; Boundary condition; Adaptive method; Uniqueness theorem; Partial differential equation; Finite element method; Drift diffusion model; Numerical simulation; M matrix; Mesh method
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/S0377-0427(03)00538-7

Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive finite element solution of semiconductor equations in terms of the Slotboom variables. The adaptive meshes are generated by the 1-irregular mesh refinement scheme. Based on these unstructured meshes and a corresponding modification of the Scharfetter–Gummel discretization scheme, it is shown that the resulting finite element stiffness matrix is an M-matrix which together with the Shockley–Read–Hall model for the generation–recombination rate leads to an existence–uniqueness–comparison theorem with simple upper and lower solutions as initial iterates. Numerical results of simulations on a MOSFET device model are given to illustrate the accuracy and efficiency of the adaptive and monotone properties of the present methods.