A Comparison of Iterative Methods for a Model Coupled System of Elliptic Equations

Many interesting areas of current industry work deal with non-linear coupled systems of partial differential equations. We examine iterative methods for the solution of a model two-dimensional coupled system based on a linearized form of the two carrier drift-diffusion equations from semiconductor modeling. Discretizing this model system yields a large non-symmetric indefinite sparse matrix. To solve the model system various point and block methods, including the hybrid iterative method Alternate Block Factorization (ABF), are applied. We also employ GMRES with various preconditioners, including block and point incomplete LU (ILU) factorizations. The performance of these methods is compared. It is seen that the preferred ordering of the grid variables and the choice of iterative method are dependent upon the magnitudes of the coupling parameters. For this model, ABF is the most robust of the non-accelerated iterative methods. Among the preconditioners employed with GMRES, the blocked “by grid point” version of both the ILU and MILU preconditioners are the most robust and the most time efficient over the wide range of parameter values tested. This information may aid in the choice of iterative methods and preconditioners for solving more complicated, yet analogous, coupled systems.