Multigrid Methods

Abstract

Numerical modeling of important phenomena and processes in science and engineering has emerged in recent years as an alternative to the experimental approach. The ever‐increasing computer speed and storage capabilities enable modeling of large‐scale problems in continuum mechanics, electromagnetism, quantum physics, and other important areas. To perform the modeling task efficiently, sophisticated algorithms and numerical techniques for the solution of the underlying models are essential. Such models are expressed frequently as mathematical equations. To represent a continuous mathematical model on a computer, which operates on discrete binary‐coded data, a mathematical procedure referred to as the discretization needs to be deployed. A common result of the discretization procedure is a system of linear or non linear equations that needs to be solved. Numerical modeling of complex systems frequently requires the solution of large systems of equations (numbering several million). Such systems need to be solved repeatedly, as part of an iterative procedure, such as linearization, time‐stepping, optimization, or inverse problems. This reason is why the solution of linear systems is a crucial phase of the overall numerical modeling process. As a result, substantial research effort has been devoted to the development of effective numerical algorithms for the solution of linear systems. This article describes an important class of iterative algorithms for the solution of linear systems, referred to as multigrid (MG) methods. MG was developed originally in the context of the solution of discretized differential equations, with the aim to overcome the difficulties that standard iterative solvers encounter in this context. Over time, MG methods have evolved into an independent field of research, which has a major impact on many scientific and engineering disciplines. In this article, we present a brief review of MG and its applications to some realistic problems from numerical modeling practice.