Analysis of a multigrid method for solving Poisson's equation with full Neumann boundary conditions

Multigrid convergence rates of Poisson's equation on rectangular domains with Dirichlet boundary conditions are standard. These ideal rates have been regained for non-Dirichlet conditions by adopting full weighting restriction operator near boundaries. Applying Neumann-conditions along the complete boundary results in illposed problem; solvable to a constant function, if some compatibility condition is satisfied. Even when the compatibility condition is fulfilled, the discrete algebraic system is usually inconsistent. This problem is analyzed and proofs are given that the multigrid method converges with ideal rates to the level of discretization error.