Convergence of the multigrid full approximation scheme for a class of elliptic mildly nonlinear boundary value problems

Summary The multigrid full approximation scheme (FAS MG) is a well-known solver for nonlinear boundary value problems. In this paper we restrict ourselves to a class of second order elliptic mildly nonlinear problems and we give local conditions, e.g. a local Lipschitz condition on the derivative of the continuous operator, under which the FAS MG with suitably chosen parameters locally converges. We prove quantitative convergence statements and deduce explicit bounds for important quantities such as the radius of a ball of guaranteed convergence, the number of smoothings needed, the number of coarse grid corrections needed and the number of FAS MG iterations needed in a nested iteration. These bounds show well-known features of the FAS MG scheme.