The full approximation storage multigrid scheme: A 1D finite element example

TL;DR

This paper presents a detailed implementation and analysis of the full approximation storage multigrid scheme applied to a 1D finite element boundary value problem, demonstrating optimal performance and convergence.

Contribution

It introduces the FAS multigrid scheme for 1D finite element problems, including implementation details and performance analysis.

Findings

Optimal work proportional to unknowns achieved

Convergence nearly to discretization error in a single cycle

Effective use of FAS V- and F-cycles with nonlinear smoothing

Abstract

This note describes the full approximation storage (FAS) multigrid scheme for an easy one-dimensional nonlinear boundary value problem. The problem is discretized by a simple finite element (FE) scheme. We apply both FAS V-cycles and F-cycles, with a nonlinear Gauss-Seidel smoother, to solve the resulting finite-dimensional problem. The mathematics of the FAS restriction and prolongation operators, in the FE case, are explained. A self-contained Python program implements the scheme. Optimal performance, i.e. work proportional to the number of unknowns, is demonstrated for both kinds of cycles, including convergence nearly to discretization error in a single F-cycle.