Multigrid Methods for Process Simulation

It was about 1985 when both of the authors started their work using multigrid methods for process simulation problems. This happened in­ dependent from each other, with a completely different background and different intentions in mind. At this time, some important monographs appeared or have been in preparation. There are the three "classical" ones, from our point of view: the so-called "1984 Guide" [12J by Brandt, the "Multi-Grid Methods and Applications" [49J by Hackbusch and the so-called "Fundamentals" [132J by Stiiben and Trottenberg. Stiiben and Trottenberg in [132J state a "delayed acceptance, resent­ ments" with respect to multigrid algorithms. They complain: "Nevertheless, even today's situation is still unsatisfactory in several respects. If this is true for the development of standard methods, it applies all the more to the area of really difficult, complex applications." In spite of all the above mentioned publications and without ignoring important theoretical and practical improvements of multigrid, this situa­ tion has not yet changed dramatically. This statement is made under the condition that a numerical principle like multigrid is "accepted", if there exist "professional" programs for research and production purposes. "Professional" in this context stands for "solving complex technical prob­ lems in an industrial environment by a large community of users". Such a use demands not only for fast solution methods but also requires a high robustness with respect to the physical parameters of the problem.

1 Introduction.- 2 A Practical Guide to Standard Multigrid Methods.- 2.1 Continuous and Discrete Model Problems.- 2.2 The Multigrid Principle.- 2.3 The Components of the Algorithm.- 2.4 Multigrid Strategies for Model Problems—Complete Algorithms.- 2.5 The Full Approximation Scheme.- 2.6 Full Multigrid.- 2.7 Multigrid Methods for Refined Grids.- 2.8 Parabolic Initial Boundary Value Problems.- 2.9 Systems of Partial Differential Equations.- 2.10 Tools to Estimate Multigrid Convergence.- 2.11 Multigrid on Parallel Computers.- 2.12 Standard Multigrid for Semiconductor Device Simulation.- 3 Adaptive Multilevel Grid Selection Strategies for Process Simulation Evolution Problems.- 3.1 Adaptive Multilevel Discrete Approximation.- 3.2 Discretization Errors and Their Properties.- 3.3 Evaluation of the Discretization Errors by Two-Level Extrapolation Techniques.- 3.4 Local Refinement Criteria Based on Discretization Errors.- 3.5 Two-Level Time Stepping Schemes Based on Extrapolation Techniques.- 4 Tayloring Multigrid Components for a Diffusion Model Problem.- 4.1 The Physical Problem and the Mathematical Representation.- 4.2 The Discretization.- 4.3 Relaxation Methods.- 4.4 Results of the Smoothing Analysis with Respect to Time-Step Size.- 4.5 Experiments with the Crank-Nicolson Scheme and with the Fully Implicit Scheme.- 4.6 The Prolongation of Grid Functions.- 4.7 The Restriction of Approximations and of Residuais.- 4.8 Technical Aspects of Time-Dependent Refinements.- 4.9 Numerical Results for the Model Problem.- 5 Procedures for Adaptive Multigrid Simulation of Evolution Processes.- 5.1 Practical Algorithmic and Programming Techniques for Multilevel Local Grid Refinement.- 5.2 The Adaptive Construction of an Initial State Discrete Approximation.- 5.3 Basic Procedures for theAdaptive Transient Simulation of Evolution Problems.- 5.4 Local Grid-Decomposition Algorithmic Strategy for Simulation of Multiparticle Evolution Processes.- 5.5 Simulation of Critical Process Simulation Steps of BiCMOS Technology—Case Study.- References.