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.

翻译：本说明描述简单单维非线性边界值问题的全近似储存(FAS)多格格内办法。 问题由简单的有限要素(FE)办法分解。 我们采用FAS V- 周期和F- 周期, 使用非线性高斯- 赛德尔平滑器来解决由此产生的有限维问题。 FAS限制和延长操作员的数学在 FE 案中作了解释。 一个自成一体的Python 方案执行这个办法。 最佳性能, 即与未知数成比例的工作, 在两种周期中都得到了证明, 包括几乎与单一F- 周期的离散错误的趋同。