Multigrid Primer: Basic Principles

TL;DR

This primer explains the fundamental principles of multigrid methods for solving elliptic PDE discretizations, emphasizing a variational quadratic energy minimization perspective to clarify algorithm design.

Contribution

It offers a simplified, variational formulation-based exposition of multigrid methods, highlighting their principles and development as both iterative and direct solvers.

Findings

Clarifies multigrid principles in a variational setting

Describes multigrid as iterative and direct solver

Aims for discretization-level accuracy efficiently

Abstract

The goal of this primer is to provide a relatively short exposition of the basics of multigrid methods, simplified by focusing on fundamental concepts in a variational setting. This is done by way of a quadratic energy minimization formulation of symmetric positive definite linear equations arising from the discretization of elliptic partial differential equations. This focus provides an alternate viewpoint to other expositions and, more importantly, it enables a simplification of the development while clarifying the principles that lead to effective algorithms. The development begins with multigrid as an iterative solver exemplified by the so-called V-cycle. It then introduces the full multigrid method as a direct solver in the sense that it is aimed directly at the source of the matrix equations. In this way, full multigrid attempts to achieve discretization-level accuracy at a cost comparable to that of a few matrix multiplies on the finest level.