Solvers

  The linear system can be solved by direct methods or iterative methods. The most common direct method is the Gauß elimination method [Eis81]. Generally, Gauß solvers give a reliable solution for ill-conditioned systems, where iterative solvers might fail. As a drawback the higher memory consumption and solving time should be noted. This prevents direct methods to be applied for three-dimensional applications in many cases.

Iterative solvers are quite popular for process and device simulation because of their fixed memory consumption during one iteration cycle. It depends on the numerical algorithm how many iterations are necessary to finally solve the system. Efforts have been made to use successive over-relaxation (SOR) and conjugate gradient (CG) solvers for ill-conditioned linear systems. We favor a bi-conjugate gradients stabilized (BiCGStab) solver in combination with ILU pre-conditioning to solve the diffusion equations. Details on the linear solving algorithms can be found in [vdV92] [Hei92] [Ada88].