Multigrid Methods for Process Simulation

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.