The author presents the monotone including multigrid algorithm MMG2 for solving weakly nonlinear elliptic boundary value problems of the form \(Lu=f(x,u),\quad x\in \Omega,\) \(u=0,\quad x\in \partial \Omega,\) where \(\Omega \subset R^ 2\) is an open domain with sufficiently smooth boundary, L is a uniformly strong elliptic differential operator of the second kind and f: \(\Omega\times R\to R\) a continuous function. The proposed MMG2 algorithm is based on the monotone multigrid algorithm (MMG) of \textit{Zou Jun} [J. Comput. Math. 5, 325-335 (1987; Zbl 0648.65069)], but is faster than it. The monotonicity of the algorithm is obtained by multiplying the interpolated coarse-grid correction with componentwise adapted parameters. Two numerical examples are also given in the paper.