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Massively parallel methods for semiconductor device modelling. (English) Zbl 0861.65111

Authors’ abstract: We describe, analyse and implement a parallel iterative method for the solution of the steady-state drift diffusion equations governing the behaviour of a semiconductor device in two space dimensions. The unknows in our model are the electrostatic potential and the electron and hole quasi-Fermi potentials. Our discretisation consists of a finite element method with mass lumping for the electrostatic potential equation and a hybrid finite element with local current conservation properties for the continuity equations.
A version of Gummel’s decoupling algorithm which only requires the solution of positive definite symmetric linear systems is used to solve the resulting nonlinear equations. We show that this method has an overall rate of convergence which only degrades logarithmically as the mesh is refined. Indeed the (inner) nonlinear solvers of the electrostatic potential equation converge quadratically, with a mesh independent asymptotic constant. We also describe an implementation on a MasPar MP-1 data parallel machine, where the required linear systems are solved by the preconditioned conjugate gradient method.
Domain decomposition methods are used to parallelise the required matrix-vector multiplications and to build preconditioners for these very poorly-conditioned systems. Our preconditioned linear solvers also have a rate of convergence which degrades logarithmically as the grid is refined relative to subdomain size, and their performance is resilient to the severe layers which arise in the coefficients of the underlying elliptic operators. Parallel experiments are given.
Reviewer: J.Mandel (Denver)

MSC:

65Z05 Applications to the sciences
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
78A55 Technical applications of optics and electromagnetic theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65H10 Numerical computation of solutions to systems of equations
65Y05 Parallel numerical computation

Software:

PLTMG
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References:

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