# Multigrid methods: from geometrical to algebraic versions

• Gundolf Haase
• Ulrich Langer
Chapter
Part of the NATO Science Series book series (NAII, volume 75)

## Abstract

Nowadays the multigrid technique is one of the most efficient methods for solving a large class of problems including elliptic boundary value problems for partial differential equations (PDEs) or systems of PDEs. Our lecture notes start with a motivation for the multigrid idea and a brief review of the multigrid history, its situation and the perspectives. The next part of our paper is devoted to the algorithmical aspects of the construction of multigrid methods including also algebraic approaches and parallelization techniques. Algebraic multigrid methods are now quite popular in practical applications because they can be included in conventional finite element packages without changing the data structure of the package. In the theoretical part of our lecture, we present some general approaches to the convergence and efficiency analysis of multigrid methods. Finally, we discuss implementation issues and present some numerical results obtained from the application of our algebraic multigrid package PEBBLES to large-scale problems in medicine and engineering.

## Keywords

Coarse Grid Fine Grid Multigrid Method Finite Element Equation Multigrid Algorithm
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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