Introduction

Multigrid methods can be used to efficiently solve elliptic partial differential equations, such as Poisson's equation. The error in the solution is iteratively damped on a hierarchy of grids, with the coarse grids reducing the low frequency (i.e., long wavelength) error components, and the fine grids the high frequency components.

For an introduction to multigrid methods, consult for example [3] or [10]. The multigrid implementation in Afivo is described in [9]. Here is a brief summary:

Afivo uses FAS (Full Approximation Scheme) multigrid. This means that the (approximate) solution is available on all refinement levels
A basic V-cycle and FMG-cycle are implemented
A procedure for the conservative filling of ghost cells is implemented
There is built-in support for Poisson's/Laplace's equation in 2D, 3D and cylindrical coordinates
Gauss-Seidel red-black smoothers are implemented
The user can customize the elliptic operator as well as the prolongation, restriction, and correction method.

How to use multigrid in Afivo

See for example one of the following examples: