I am applying the adaptive mesh in an elliptic equation, and when I use Neumann conditions for one linear or quadratic equation, the convergence is reached, but when the trigonometric equations is used the maximum residual doesn't reach the tolerance that I want (e10-8), or be, it stall. I am thinking that the problem can be in the order of the polynomials that I am using to do the interpolations between the coarse and fine mesh for the ghost cells or in the restriction and in the prolongation of the multigrid method. Will I have a good behavior for this case.?

My experience with multigrid for a equation with Neumann boundary condions, care is needed for the corresponding Neumann conditions on the coarse grid. The incremental form of the Neumann conditions is desirable, rather than the original form.

but when you said "The incremental form of the Neumann" are you talking about the third order in the discretization of the neumann condition??? because I am using the second order.

The initial boundary values Q_b on a coarse grid is restricted from the next finer grid, now you are going to iterate Q as well as Q_b. It is suggested to solve LHS * Delta Q = RHS on the coarse grid, where Delta Q is itearted incremental form. Once Delta Q is obtained, Q_i=Q_i^n+ Delta Q_i+1 (first order accuracy), if i=1 is the boundary point.