Coupled and decoupled algorithms for semiconductor simulation

Algorithms for the numerical simulation are analyzed by computers of the steady state behavior of MOSFETs. The discretization and linearization of the nonlinear partial differential equations as well as the solution of the linearized systems are treated systematically. Thus we generate equations which do not exceed the floating point representations of modern computers and for which charge is conserved while appropriate maximum principles are preserved. A typical decoupling algorithm of the solution of the system of pde is analyzed as a fixed point mapping T. Bounds exist on the components of the solution and for sufficiently regular boundary geometries higher regularity of the derivatives as well. T is a contraction for sufficiently small variation of the boundary data. It therefore follows that under those conditions the decoupling algorithm coverges to a unique fixed point which is the weak solution to the system of pdes in divergence form. A discrete algorithm which corresponds to a possible computer code is shown to converge if the discretizaion of the pde preserves the regularity properties mentioned above. A stronger convergence result is obtained by employing the higher regularity for enforcing the weak formulations of the pde more strongly. The execution speed of a modification of Newton's method, two versions of a decoupling approach and a new mixed solution algorithm are compared for a range of problems. The asymptotic complexity of the solution of the linear systems is identical for these approaches in the context of sparse direct solvers if the ordering is done in an optimal way.